\begin{theorem}
\label{theo:main}
For every non-orientable regular map $\M$ of type $\{p,q\}$ where at least one of $p$ or $q$ is odd, there exists an unstable fully transitive $2$-orbit map $\M^\omega$, that covers $\M$. Moreover, for all $n>3$ there exist $2^{n-3}$ non-isomorphic unstable $2$-orbit fully transitive $n$-maniplexes, whose $3$-faces are isomorphic to $\M^\omega$.
\end{theorem}

Every non-bipartite graph $\Gamma$ admits a connected {\em canonical double cover} $\widetilde{\Gamma}$, isomorphic to the direct graph product $\Gamma \times K_2$. The group of automorphisms of $\widetilde{\Gamma}$ contains a subgroup isomorphic to $\hbox{\rm Aut}(\Gamma) \times \ZZ_2$, but may be, in general, significantly larger. We say $\Gamma$ is stable precisely when $\hbox{\rm Aut}(\widetilde{\Gamma}) \cong \hbox{\rm Aut}(\Gamma) \times \ZZ_2$, and say it is unstable otherwise.
Stability in graphs was first studied in \cite{Marusic1989} and have since received considerable attention (see for instance, \cite{BlasDoubleCovers,ademirdorde,QIN2019154,Surowski,Surowski2001Stability,WILSON2008359}).

The problem of stability extends naturally to other types of combinatorial objects, that can ultimately be regarded as graphs. For instance, in \cite{gareth} Jones considers stability in maps on surfaces. Typically, a map is defined as the embedding of a connected graph on a compact surface with no boundary, with the property that the embedded graph divides the surface into simply connected regions. Every map uniquely determines an edge-coloured graph, called its {\em flag-graph}, that completely encodes its structure, allowing us to define maps in pure graph-theoretical terms. What is more, a map $\widetilde{M}$ is the canonical double cover of a map $\M$, in the topological sense, if and only if the flag graph of $\widetilde{M}$ is the canonical double cover of the flag graph of $\M$, in the graph theoretical sense. That is, the problem of stability in maps connects naturally to stability in graphs. There is however, a caveat. While there is a correspondence between the automorphism of a map and those of its flag graph, this correspondence is, in general, not one-to-one. The automorphisms of a map $\M$ correspond precisely to the edge-colour preserving automorphism of its flag graph, and only to those. As a consequence of this, the group of automorphism of a map, when seen as a flag-graph, acts semiregularly on its nodes, which are usually called {\em flags} in this context. This implies, as Jones remarks in \cite{gareth}, that every regular (maximally symmetric) non-orientable map is stable. Therefore, the problem of stability takes a somewhat different flavour in the setting of maps, as stable maps are very easily produced. A natural and more interesting question is whether weaker symmetry conditions also imply stability, which is partially answered in \cite{gareth}. The degree of symmetry of a map is usually measured by the number of flag-orbits of its automorphism group. Maps whose automorphism group has exactly $k$ distinct flag-orbits are called $k$-orbit maps, with $1$-orbit maps (which we call {\em regular} here) being by far the most studied class.
Jones proves, among other things, that unstable edge-transitive maps exist. However, with the exception of a small degenerate map, having (one face, one vertex and two edges). all the maps constructed in \cite{gareth} have $4$ or more orbits on flags. The question whether $2$-orbit maps exist is left unanswered.

In this paper, we consider this question in the much broader setting of maniplexes (\cite{wilson2012maniplexes}). A maniplex of rank $n$ is a properly $n$-edge-coloured $n$-valent connected graph that generalises the notion of a flag-graph. In particular, a maniplex of rank $3$ is the combinatorial (graph-theoretical) equivalent of a map on a surface, in the sense that the flag-graph of a map is a $3$-maniplex, and every $3$-maniplex arises in this way. In short, maniplexes can be regarded as higher-dimension analogues of maps on surfaces (or as graph-theoretical generalisations of abstract polytopes \cite{mcmullen2002abstract}, \cite{garza2018polytopality}).
Naturally, many of the notions (such as symmetry or orientability) of the theory of maps extend to maniplexes of higher ranks in an intuitive way. We call the nodes of a maniplex flags and we define a maniplex automorphism as an edge-colour preserving graph automorphism.
We say a maniplex is regular (often also called reflexible) if its automorphism group is transitive on the flags.
Just as with maps, every regular non-orientable maniplex is automatically stable and thus the problem of finding highly symmetric (but non-regular) unstable maniplexes is natural. In this paper, we prove that for rank $n\geq3$, there exist infinitely many $2$-orbit $n$-maniplexes that are unstable. Moreover, the maniplexes constructed are fully-transitive, meaning that their group of automorphisms is transitive on the set of faces of rank $i$ for every $i<n$. Precise definitions of the terms appearing in these paragraphs are provided in \Cref{sec:preliminaries} and throughout the paper. We now state the main theorem of this paper.

In this paper, we consider this question in the much broader setting of maniplexes (\cite{wilson2012maniplexes}). A maniplex of rank $n$ is a properly $n$-edge-coloured $n$-valent connected graph that generalises the notion of a flag-graph. In particular, a maniplex of rank $3$ is the combinatorial (graph-theoretical) equivalent of a map on a surface, in the sense that the flag-graph of a map is a $3$-maniplex, and every $3$-maniplex arises in this way. In short, maniplexes can be regarded as higher-dimension analogues of maps on surfaces (or as graph-theoretical generalisations of abstract polytopes \cite{mcmullen2002abstract}, \cite{garza2018polytopality}).
Naturally, many of the notions (such as symmetry or orientability) of the theory of maps extend to maniplexes of higher ranks in an intuitive way. We call the nodes of a maniplex flags and we define a maniplex automorphism as an edge-colour preserving graph automorphism.
We say a maniplex is regular (often also called reflexible) if its automorphism group is transitive on the flags.
Just as with maps, every regular non-orientable maniplex is automatically stable and thus the problem of finding highly symmetric (but non-regular) unstable maniplexes is natural. In this paper, we prove that for rank $n\geq3$, there exist infinitely many $2$-orbit $n$-maniplexes that are unstable. Moreover, the maniplexes constructed are fully-transitive, meaning that their group of automorphisms is transitive on the set of faces of rank $i$ for every $i<n$. Precise definitions of the terms appearing in these paragraphs are provided in \Cref{sec:preliminaries} and throughout the paper. We now state the main theorem of this paper.

The basis for our construction consists of two operations: the cross-cover operation for graphs, first defined in \cite{WILSON2008359}, and the colour-coded extensions for maniplexes (see \cite{twistop}), which we define here in graph-theoretical terms.

For $i \in \nset$ we say $\M$ is $i$-face-transitive if $\aut(\M)$ acts transitively on the set of $i$-faces of $\M$. If $\M$ is $i$-face-transitive for all $i \in \nset$ then we say $\M$ is \emph{fully transitive}. Note that information about the $i$-face-transitivity of $\M$ is encoded in its symmetry-type graph. Indeed, the number of orbits of $\Aut(\M)$ on $i$-faces is equal to the number of connected components of the multigraph resulting from deleting all edges and semi-edges of colour $i$ from $\stg(\M)$. Regular maniplexes are fully-transitive, although not all fully-transitive maniplexes are regular. Figure \ref{fig:2orbstgs} shows the seven possible symmetry-type graphs for a $2$-orbit $3$-maniplex. Observe that those on the top row correspond to fully-transitive maniplexes, while those on the bottom correspond to maniplexes that are not $i$-face-transitive for some $i$. Thus, we can think of a $2$-orbit maniplex of the first kind as being 'more symmetric' than those of the second kind.

Now, consider the graph $Q_\ell$ whose vertices are the facets of $\M$, where two vertices are adjacent if the corresponding facets are joined by an $n$-edge in $\M$ (that is, if there in an $n$-edge having one endpoint in the first facet, and the other in the second facet). The graph $Q_\ell$ is isomorphic to the $\ell$-cube graph and every automorphism of $\M$ (which permutes its facets) induces a permutation of the vertices of $Q_\ell$. Moreover, since an automorphism of $\M$ maps $n$-edges to $n$-edges, the induced permutation on the vertices of $Q_\ell$ must map adjacent vertices to adjacent vertices. In other words, every automorphism of $\M$ induces a graph automorphism of $Q_\ell$. Let $G = \Aut(\M)$ and let $G'\leq\Aut(Q_\ell)$ be the group of automorphisms of $Q_\ell$ induced by $G$. Now consider the group $H = \langle \tau_1,\ldots,\tau_\ell \rangle$ and observe that $H$ acts not only transitively, but regularly, on the set of facets of $\M$. It follows that the group $H'$ of automorphisms of $Q_\ell$ induced by $H$ acts regularly on the vertices of $Q_\ell$. Moreover, one can see that the set of all pairs of antipodal vertices of $Q_\ell$ is a block system for $H'$ (indeed, $H'$ acts on the main diagonals of the $\ell$-cube). Since this is also a block system for the full automorphism group of $Q_\ell$, and $H' \leq G' \leq \Aut(Q_\ell)$, we see that it is a block system for $G'$ too. This block system corresponds to the partition into chromatic classes (of the facets of $\M$) with the antipodal colouring $\C^\times$ of $\M$, and thus, this partition must be a block system for $\Aut(\M)$ (acting on facets). We conclude that the antipodal colouring is $\Aut(\M)$-invariant.
\end{proof}

\Cref{prop:pares} tells us that it suffices to find a map $\M$ with a weight function $\omega$ such that $\M^\omega$ is a non-orientable maniplex (which must necessarily be unstable by \Cref{theo:cross}), to show that there are unstable maniplexes of every rank $n \geq 3$. Finding such a pair $(\M,\omega)$ is in fact not difficult, as we will see in the following pages. However, the maniplexes obtained by extending $\M$ and $\omega$ may, in principle, be of any symmetry type. In fact, they may very well completely lack symmetry. Since we are interested in maniplexes that are both unstable and very symmetric, extra conditions need to be imposed when considering a pair $(\M,\omega)$. This motivates the following definition.

\begin{lemma}
\label{lem:nicecover}
Let $\M$ be a regular $n$-maniplex and let $\omega\colon \M \to \ZZ_k$ be a weight function for $\M$. If $(\M,\omega)$ is a proper pair and $k=4$,
then $\M^\omega$ is an unstable $2$-orbit maniplex.
\end{lemma}

Theorem \ref{theo:main} now follows from Proposition \ref{prop: goodweight} and Lemmas \ref{lem:niceext}, \ref{lem:nicecover} and \ref{lem:stgcover}. Indeed, given a non-orientable regular map $\M$ of type $\{p,q\}$, where $p$ or $q$ is odd, the pair $(\M,\vartheta)$ is a proper pair by \Cref{prop: goodweight}. The cross-cover of $(\M,\vartheta)$ is an unstable $2$-orbit maniplex by \Cref{lem:nicecover}, and it has symmetry-type $2^3_{\{1\}}$ by \Cref{lem:stgcover}, making it fully-transitive.

The basis for our construction consists of two operations: the cross-cover operation for graphs, first defined in \cite{WILSON2008359}, and the colour-coded extensions for maniplexes (see \cite{twistop}), which we define here in graph-theoretical terms.

The paper is structured as follows. In \Cref{sec:preliminaries}, we give basic definitions pertaining to graphs and maniplexes. In \Cref{sec:cross} we introduce cross-covers of graph (and maniplexes) and prove a few auxiliary results relating to their connectivity and symmetry.
In \Cref{sec:ext} we discuss colour-coded extensions and define two well-known ways of extending maniplexes that will be useful for our purposes.
In \Cref{sec:extweight} we show how the notions of cross-covers and colour-coded extensions interweave nicely, and we can use a combination of both operations to produce unstable maniplexes with desirable symmetry properties.
In particular, we prove the two propositions that will be the key to the proof of \Cref{theo:main}, which we complete in \Cref{sec:proof}.

The proof of \Cref{theo:main} is based on combining two operations: the "cross-cover" operation for graphs and "colour-coded extensions" for maniplexes (defined in graph-theoretical terms). The argument develops auxiliary results on connectivity and symmetry for cross-covers, then introduces two standard extension methods, and then shows in \Cref{sec:extweight} how cross-covers and colour-coded extensions "interweave" so that a combination of both "produce unstable maniplexes with desirable symmetry properties". In particular, the proof relies on "two propositions" proved in \Cref{sec:extweight} that are "key to the proof of \Cref{theo:main}", with the theorem then completed in \Cref{sec:proof}.

In establishing the main theorem, the argument is based on combining two operations: the “cross-cover” operation for graphs and “colour-coded extensions” for maniplexes (defined in graph-theoretical terms). The argument develops auxiliary results on connectivity and symmetry for cross-covers, then introduces two standard extension methods, and then shows in \Cref{sec:extweight} how cross-covers and colour-coded extensions “interweave” so that a combination of both “produce unstable maniplexes with desirable symmetry properties”. In particular, the proof relies on “two propositions” proved in \Cref{sec:extweight} that are key to the proof of the main theorem, with the main theorem then completed in \Cref{sec:proof}.

Question: Let \(\mathcal M\) be a non-orientable regular map of type \(\{p,q\}\), meaning that every face has length \(p\), every vertex has valence \(q\), and the automorphism group of \(\mathcal M\) is transitive on flags. Assume that at least one of \(p\) or \(q\) is odd. A map or maniplex is called \(2\)-orbit if its automorphism group has exactly two flag-orbits, and fully transitive if its automorphism group is transitive on faces of every rank \(i<n\). It is called unstable if the automorphism group of its canonical double cover is larger than the expected subgroup \(\operatorname{Aut}(X)\times \mathbb Z_2\). Which statement holds under these assumptions?

There exists an unstable fully transitive \(2\)-orbit map \(\mathcal M^{\omega}\) that covers \(\mathcal M\). Moreover, for every \(n>3\), there exist \(2^{n-3}\) pairwise non-isomorphic unstable fully transitive \(2\)-orbit \(n\)-maniplexes whose \(3\)-faces are isomorphic to \(\mathcal M^{\omega}\).

Choices:
B. There exists an unstable fully transitive \(2\)-orbit map \(\mathcal M^{\omega}\) that covers \(\mathcal M\). Moreover, for every \(n\ge 3\), there exist \(2^{n-2}\) pairwise non-isomorphic unstable fully transitive \(2\)-orbit \(n\)-maniplexes whose \(3\)-faces are isomorphic to \(\mathcal M^{\omega}\).
C. There exists an unstable \(2\)-orbit map \(\mathcal M^{\omega}\) that covers \(\mathcal M\). Moreover, for every \(n>3\), there exist \(2^{n-3}\) pairwise non-isomorphic unstable \(2\)-orbit \(n\)-maniplexes whose \(3\)-faces are isomorphic to \(\mathcal M^{\omega}\).
D. There exists an unstable fully transitive \(2\)-orbit map \(\mathcal M^{\omega}\) that covers \(\mathcal M\). Moreover, for every \(n>3\), for each such \(n\) there exists at least one unstable fully transitive \(2\)-orbit \(n\)-maniplex whose \(3\)-faces are isomorphic to \(\mathcal M^{\omega}\), and these maniplexes may be chosen canonically from \(\mathcal M\), independently of the choice of \(\omega\).
E. There exists a fully transitive \(2\)-orbit map \(\mathcal M^{\omega}\) that covers \(\mathcal M\) and is stable. Moreover, for every \(n>3\), there exist \(2^{n-3}\) pairwise non-isomorphic fully transitive \(2\)-orbit \(n\)-maniplexes whose \(3\)-faces are isomorphic to \(\mathcal M^{\omega}\).

dependence on the chosen weight function \(\omega\) and exact counting conclusion

cross-cover produces instability rather than stability


\label{thm:main}
Let $G_{1}$,~$G_{2}$, \dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let
\[
W = G_{k} \operatorname{wr} G_{k-1} \operatorname{wr} \dots \operatorname{wr} G_{1}
\]
be the iterated wreath product constructed via the regular action of each factor. Set
\[
A = (G_{k}/G_{k}') \times (G_{k-1}/G_{k-1}') \times \dots \times (G_{2}/G_{2}').
\]
Then
\[
d(W)=\max_p(d(A\times G_1), d(A)+1, d_p(A)+2),
\]
where $p$ ranges over the set of prime numbers dividing both $\mathopen{|}A\mathclose{|}$ and $\mathopen{|}S\mathclose{|}$.

A classic theme of group theory research is the generation property of groups. The structure of a group can be described using its generators. There have been numerous research on generators of specific groups. For example, it is delivered in undergraduate group theory courses that symmetric groups can be generated with two elements. Using the Classification of Finite Simple Groups, it is proved that every finite non-abelian simple group is $2$\nobreakdash- generated.

One of the interesting generation problems is the growth rate of the minimum number of generators of the direct product $G^n$ where $G$ is a finite group. Wiegold shows in \cite{Wiegold} that the number of required generators grows linearly when $G$ is an imperfect group and it grows logarithmically when $G$ is perfect. Apart from direct products, another way to construct groups is via wreath products. In this paper, we will investigate the minimum number of generators of iterated wreath products.

Bhattacharjee \cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\nobreakdash- generated. Quick \cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\times G_2/G_2'\times\dots\times G_n/G_n'$ is bounded as well.

Recently, the author of this paper \cite{LuQuick} considers the generators required for the iterated wreath products of symmetric groups, alternating groups or cyclic groups. In particular, it is established that the minimum number of generators of iterated wreath products of symmetric groups and alternating groups is the maximum of $2$ and the number of symmetric groups involved. The method used relies on Lucchini's work \cite{Lu97}. Lucchini determines the number of generators required for the wreath product $A\operatorname{wr} S_n$ of an abelian group $A$ constructed via the natural action of the symmetric group of degree $n$. This is then adapted for the wreath product $A\operatorname{wr} A_n$ constructed via the natural action of the alternating group of degree $n$ in \cite{LuQuick}. In \cite{Lu97}, Lucchini also establishes the minimum number of generators for the wreath product $A\operatorname{wr} G$ constructed via the regular action of a simple group $G$. We will use this result as part of our argument and generalise it to apply to the case that $G$ is an almost simple group.

In this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\nobreakdash- generated.

To state the theorem, we recall some basic notations. For a group $G$, we write $d(G)$ for the minimum number of generators for $G$. For a finite abelian group $A$ and a prime number $p\mid\mathopen{|}A\mathclose{|}$, we write $d_p(A)$ for the minimum number of generators for the Sylow-$p$ subgroup of $A$. Because of the Fundamental Theorem of Finite Abelian Groups, $d(A)=\max_{p\mid\mathopen{|}A\mathclose{|}}d_p(A)$. We will establish the following theorem.

\begin{abstract}
 Let $G_{1}$,~$G_{2}$, \dots\ be a sequence of almost simple groups and construct a sequence~$(W_{i})$ of wreath products via $W_{1} =
 G_{1}$ and, for each $i >1$, $W_{i+1} = G_{i+1} \wr W_{i}$ via
 the regular action of each $G_i$. We determine the minimum
 number~$d(W_{i})$ of generators required for each wreath product in
 this sequence.
\end{abstract}

Bhattacharjee \cite{Bhatt} investigates the generation property of iterated wreath product of alternating groups constructed via the natural action. She proves that the probability that such products can be generated by two random elements is positive. This shows that iterated wreath products of alternating groups are $2$\nbd generated. Quick \cite{MRQ} generalises this result and shows that the same result stills holds for iterated wreath products of non-abelian simple groups constructed from any faithful transitive action of the factors. Bondarenko \cite{Bond} considers iterated wreath products of transitive permutation groups with uniformly bounded number of generators. He establishes that the number of generators of the product involving $G_1, \dots, G_n$ is bounded if and only if the number of generators of the abelian direct product $G_1/G_1'\times G_2/G_2'\times\dots\times G_n/G_n'$ is bounded as well.

In this article, we will establish the precise number of generators required for an iterated wreath product constructed from the regular action of almost simple groups. If we take the factors involved to be simple groups, then we recover the observation that iterated wreath products constructed via the regular action of non-abelian simple groups are $2$\nbd generated.

We finish this introduction by giving the structure of this paper. In Section~\ref{sec:pre}, we recall basic definition that we will use. We will also introduce the results that our conclusions depend on, particularly the ones from \cite{Lu97}. In Section~\ref{sec:AwrG}, using the results in~\cite{Lu97} and also extending them to cover a more general case, we will determine the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action. We will then provide the main proof of the theorem in the final section. We will prove it by showing that $d(W)=d(A\wr G_1)$ and determining $d(A\wr G_1)$, where $W, A, G_1$ are as defined in Theorem~\ref{thm:main} (see Lemmas~\ref{lem:keylemmaregular-abelian} and \ref{lem:regularinduction} respectively).

\begin{lem}\label{lem:LuStam}
\begin{enumerate}
\item\label{i:Lucchini-s}{\normalfont (Lucchini~{\cite[1.2]{Lu97}})}
 Let $M$~be an irreducible $G$\nbd module for a finite group~$G$.
 Then
 \[
 s_{G}(M) = \delta_{G}(M) + \dim_{\End{M}} \Cohom{G/\Cent{G}{M},M}.
 \]
\item\label{i:H1<rGM}{\normalfont (Lucchini~{\cite[1.3]{Lu97}})}
Let $M$ be an irreducible $G$\nbd module. Then
\[
\dim_{\End{M}}H^1(G/C_G(M),M)<r_G(M).
\]
\item\label{i:Stam}{\normalfont (Stammbach~{\cite[Theorem A]{Stammbach}})}
Given a prime integer $p$, a finite group $G$ is $p$-soluble if and only if $\Cohom{G/C_G(M),M}=0$ for all irreducible $\Field{p}G$\nbd modules $M$.
\item\label{i:augmentationgenerator}{\normalfont (Lucchini~{\cite[Proposition 1]{Lu97}})}
Let $H$ be a finite group and $G$ be a transitive permutation group of degree $n$, then
\begin{align*}
d(I_{H\wr G})=\max\left(d(I_{(H/H')\wr G}), \left\lfloor\frac{d(I_H)-2}{n}\right\rfloor+2\right).
\end{align*}
\item\label{i:soluble}{\normalfont (Lucchini~{\cite[Theorem 2]{Lu97}})}
If $H$ is a finite soluble group and $G$ is a transitive permutation group of degree $n$, then
\[
d(H\wr G)=\max\left(d\left(\frac{H}{H'}\wr G\right),\left\lfloor\frac{d(H)-2}{n}\right\rfloor+2\right).
\]
\item\label{i:Lucchini-AwrG}{\normalfont (Lucchini~{\cite[Theorem 4]{Lu97}})}
Let $A\wr G$ be the wreath product of a finite abelian group $A$ by a finite group $G$ constructed via the regular action of $G$. Set
\[
\rho_p=\max_Mh_G(M)+d_p(A)
\]
where $M$ ranges over the set of non-trivial irreducible $\Field{p}G$\nbd modules, with $\rho_p=0$ if every irreducible $\Field{p}G$\nbd modules is trivial. Then
\[
d(A\wr G)=\max_{p\mid\order{A}}(d(A\times G),\rho_p).
\]
\end{enumerate}
\end{lem}

\begin{lem}\label{lem:AwrGLietype}
Let $S$ be a simple group of Lie type and $G$ be an almost simple group with socle $S$. Let $A$ be a finite abelian group and set $W=A\wr G$ to be the wreath product of the abelian group by the almost simple group with respect to the regular action. Then
\[
d(W)=\max_q(d(A\times G), d(A)+1, d_q(A)+2),
\]
where $q$ ranges over the set of the prime numbers dividing $\order{A}$ and $\order{S}$.
\end{lem}
\begin{prf}
Using Lemma~\ref{lem:LuStam}\ref{i:Lucchini-AwrG}, it suffices to show that
\begin{equation}\label{eq:maxsigma=dq}
\max_{p\mid\order{A}}\rho_p=\max_q(d(A)+1, d_q(A)+2),
\end{equation}
where $q$ ranges over the set of the prime numbers dividing $\order{A}$ and $\order{S}$ and
\[
\rho_p=\max_Mh_G(M)+d_p(A).
\]
Lemma~\ref{lem:maxhG} tells us that $\rho_p\geq1+d_p(A) $ for all $p\mid\order{A}$. Hence $\rho_p\geq 1+d(A)$. Lemma~\ref{lem:maxhG} also shows that, for any prime number $p$ dividing $\order{A}$ and $\order{S}$, $\rho_p=2+d_p(A)$. Thus
\[
\max_{p\mid\order{A}}\rho_p\geq\max_{p\mid\order{A}, p\mid\order{S}}\rho_p=\max_{p\mid\order{A}, p\mid\order{S}} 2+d_p(A).
\]
Then
\[
\max_{p\mid\order{A}}\rho_p\geq\max_q(d(A)+1, d_q(A)+2).
\]

We will now establish the main theorem. Accordingly, let $k\geq2$, and let $G_1, G_2,\dots, G_k$ be almost simple groups. Define
\[
W=G_k\wr G_{k-1}\wr\dots\wr G_1
\]
to be the wreath product constructed via the regular action of each factor. We also define
\[
H=G_k\wr G_{k-1}\wr \dots\wr G_2
\]
and set $A=H/H'$ to be the abelianization of $H$; that is,
\[
A=(G_k/G_k')\times (G_{k-1}/G_{k-1}')\times\dots\times (G_{2}/G_{2}').
\]

\begin{lem}\label{lem:keylemmaregular-abelian}
Let $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\wr G_1$ is given by
\[
d(A\wr G)=\max_q(d(A\times G), d(A)+1, d_q(A)+2),
\]
where $q$ ranges over the set of prime numbers dividing $\order{A}$ and $\order{S}$.
\end{lem}
\begin{prf}
The case when $S$ is a simple group of Lie type is given in Lemma~\ref{lem:AwrGLietype}. If $S$ is an alternating group, then $\Out(S)$ is abelian. According to~\cite[Table 1]{Atlas}, $\Out(S)$ is also abelian when $S$ is one of the sporadic groups. It follows that the abelian chief factors of $G$ are all trivial in the case that $S$ is an alternating simple group or one of the sporadic simple groups. Then for any non-trivial $\Field{p}G$\nbd module $M$, $\delta_G(M)=0$. In this case, the formula is a direct consequence of~\cite[Corollary 5]{Lu97}.
\end{prf}

\begin{lem}\label{lem:regularinduction}
  Let $W$~and~$A$ be as defined above. Then
  \begin{equation}
    d(W) = d(A \wr G_{1}).
    \label{eq:regularreduction}
  \end{equation}
\end{lem}

\begin{thm}\label{thm:main}
Let $G_{1}$,~$G_{2}$, \dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let
\[
W = G_{k} \wr G_{k-1} \wr \dots \wr G_{1}
\]
be the iterated wreath product constructed via the regular action of each factor. Set
\[
A = (G_{k}/G_{k}') \times (G_{k-1}/G_{k-1}') \times \dots \times (G_{2}/G_{2}').
\]
Then
\[
d(W)=\max_p(d(A\times G_1), d(A)+1, d_p(A)+2),
\]
where $p$ ranges over the set of prime numbers dividing both $\order{A}$ and $\order{S}$.
\end{thm}

We finish this introduction by giving the structure of this paper. In Section~\ref{sec:pre}, we recall basic definition that we will use. We will also introduce the results that our conclusions depend on, particularly the ones from \cite{Lu97}. In Section~\ref{sec:AwrG}, using the results in~\cite{Lu97} and also extending them to cover a more general case, we will determine the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action. We will then provide the main proof of the theorem in the final section. We will prove it by showing that $d(W)=d(A\operatorname{wr} G_1)$ and determining $d(A\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in Theorem~\ref{thm:main} (see Lemmas~\ref{lem:keylemmaregular-abelian} and \ref{lem:regularinduction} respectively).

\paragraph{Acknowledgements:} The author is funded by the China
Scholarship Council.

To prove Theorem~\ref{thm:main}, the paper "determin[es] the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action" (using results in~\cite{Lu97} and extending them). The "main proof" is then given by "showing that $d(W)=d(A\operatorname{wr} G_1)$ and determining $d(A\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in Theorem~\ref{thm:main}" (see Lemmas~\ref{lem:keylemmaregular-abelian} and~\ref{lem:regularinduction}).

To prove the main theorem, the paper determin[es] the minimum number of generators of the wreath product of an abelian group by an almost simple group constructed via the regular action (using results in~\cite{Lu97} and extending them). The main proof is then given by showing that $d(W)=d(A\operatorname{wr} G_1)$ and determining $d(A\operatorname{wr} G_1)$, where $W, A, G_1$ are as defined in the statement of the main theorem. We first prove the following lemma.

\begin{lem}\label{lem:regularinduction}
  Let $W$~and~$A$ be as defined above. Then
  \begin{equation}
    d(W) = d(A \wr G_{1}).
    \label{eq:regularreduction}
  \end{equation}
\end{lem}

Next we determine $d(A\wr G_1)$ via the following lemma.

\begin{lem}\label{lem:keylemmaregular-abelian}
Let $S$ be the socle of $G_1$. The minimum number of generators of the wreath product $A\wr G_1$ is given by
\[
d(A\wr G)=\max_q(d(A\times G), d(A)+1, d_q(A)+2),
\]
where $q$ ranges over the set of prime numbers dividing $\order{A}$ and $\order{S}$.
\end{lem}

\label{thm:main}
Let $G_{1}$,~$G_{2}$, \dots,~$G_{k}$ be a sequence of almost simple groups. Let $S$ be the socle of $G_1.$ Let
\[
W = G_{k} \operatorname{wr} G_{k-1} \operatorname{wr} \dots \operatorname{wr} G_{1}
\]
be the iterated wreath product constructed via the regular action of each factor. Set
\[
A = (G_{k}/G_{k}') \times (G_{k-1}/G_{k-1}') \times \dots \times (G_{2}/G_{2}').
\]
In establishing the main theorem, we show that
\[
d(W)=\max_p(d(A\times G_1), d(A)+1, d_p(A)+2),
\]
where $p$ ranges over the set of prime numbers dividing both $\mathopen{|}A\mathclose{|}$ and $\mathopen{|}S\mathclose{|}$.,

Question:
For a finite group $X$, let $d(X)$ denote the minimum number of generators of $X$. For a finite abelian group $B$ and a prime $p\mid |B|$, let $d_p(B)$ denote the minimum number of generators of a Sylow-$p$ subgroup of $B$. Let $G_1, G_2, \dots, G_k$ be a sequence of almost simple groups, let $S$ be the socle of $G_1$, form the iterated wreath product
\[
W = G_k \operatorname{wr} G_{k-1} \operatorname{wr} \dots \operatorname{wr} G_1
\]
using the regular action of each factor, and set
\[
A=(G_k/G_k')\times (G_{k-1}/G_{k-1}')\times \dots \times (G_2/G_2').
\]
Which statement holds for every such choice of groups?

\[
d(W)=\max_p\bigl(d(A\times G_1),\ d(A)+1,\ d_p(A)+2\bigr),
\]
where $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$.

Choices:
B.
\[
d(W)=\max_p\bigl(d(A\times G_1),\ d(A)+1,\ d_p(A)+2\bigr),
\]
where $p$ ranges over all prime numbers dividing $|A|$.
C.
\[
d(W)\ge \max_p\bigl(d(A\times G_1),\ d(A)+1,\ d_p(A)+2\bigr),
\]
where $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$.
D.
\[
d(W)=\max_p\bigl(d(A\times G_1),\ d(A)+1,\ d_p(A)+1\bigr),
\]
where $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$.
E.
\[
d(W)=\max_p\bigl(d(A\times S),\ d(A)+1,\ d_p(A)+2\bigr),
\]
where $p$ ranges over the prime numbers dividing both $|A|$ and $|S|$.

prime range restricted to divisors of both |A| and |S|

exact equality replaced by the implied lower bound

reduction to A\wr G_1; replacing G_1 by its socle S inside d(\,\cdot\,\times\,\cdot\, )


Question:
Let $G_{1}, G_{2}, \dots, G_{k}$ be a sequence of almost simple groups, let $S$ be the socle of $G_{1}$, and form the iterated wreath product
\[
W=G_{k}\operatorname{wr} G_{k-1}\operatorname{wr}\dots\operatorname{wr} G_{1}
\]
using the regular action of each factor. Set
\[
A=(G_{k}/G_{k}')\times (G_{k-1}/G_{k-1}')\times\dots\times (G_{2}/G_{2}').
\]
If $d(H)$ denotes the minimum number of generators of a finite group $H$, and for a finite abelian group $A$ and a prime $p\mid |A|$, $d_p(A)$ denotes the minimum number of generators of a Sylow-$p$ subgroup of $A$, which statement holds for $d(W)$?

\[
d(W)=\max_{p}\bigl(d(A\times G_{1}),\ d(A)+1,\ d_p(A)+2\bigr),
\]
where $p$ ranges over the set of prime numbers dividing both $|A|$ and $|S|$.

Choices:
B.
\[
d(W)=\max_{p}\bigl(d(A\times G_{1}),\ d(A)+1,\ d_p(A)+2\bigr),
\]
where $p$ ranges over the set of prime numbers dividing $|A|$.
C.
\[
d(W)\ge \max_{p}\bigl(d(A\times G_{1}),\ d(A)+1,\ d_p(A)+2\bigr),
\]
where $p$ ranges over the set of prime numbers dividing both $|A|$ and $|S|$.
D.
\[
d(W)=\max_{p}\bigl(d(A\times G_{1}),\ d(A)+1,\ d_p(A)+1\bigr),
\]
where $p$ ranges over the set of prime numbers dividing both $|A|$ and $|S|$.
E.
\[
d(W)=d(A\times G_{1}).
\]

prime-indexing restricted to primes dividing both |A| and |S|

exact equality replaced by lower bound only

additive constant in the Sylow term d_p(A)+2

omission of the extra wreath-product contributions d(A)+1 and d_p(A)+2 after reduction to A\wr G_1


\label{conjecture:main}
Let $\alg{A}$ and $\alg{B}$ be finite modules. Then, there are only finitely many clonoids from $\alg{A}$ to $\alg{B}$, if and only if $\alg{A}$ and $\alg{B}$ are of coprime order.

\begin{theorem} \label{theorem:main}
Let $\F$ be a finite field, $k \in \N$, and let $\algB$ be an $\algR$-module, such that $|F|$ is invertible in $\algR$. Then $\clodO_{\F^k,\algB}$ is uniformly generated by $k$-ary $(\F^k,\algB)$-minors.
\end{theorem}

A \emph{clone} is a set of finitary operations on a given set that contains all projections and is closed under functional composition. Clones are a fundamental object in universal algebra, as they describe the set of term operations of any algebraic structure. Thus, starting with Post's classification of clones on a two-element set \cite{post-lattice}, there has been a rich history of structural results on finite algebras based on clone theory, see e.g. \cite{szendrei-clones}. On the other hand, clones theory has also several applications in theoretical computer science. One of the most prominent examples is the study of polymorphism clones to determining the complexity of fixed template constraint satisfaction problems (see e.g. the survey \cite{BKW-polymorphisms}), which lead to Bulatov's and Zhuk's independent proofs of the CSP dichotomy theorem \cite{bulatov-dichotomy}, \cite{zhuk-dichotomy-short,zhuk-dichotomy}.

In recent years, generalizations of clones such as \emph{minions} and \emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \cite{BBKO-PCSP}. A \emph{clonoid} from an algebra $\alg{A}$ to an algebra $\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\alg{A}$ and (post-)composition with the operations of $\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \cite{ CF-clonoids}.

Due to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\alg{A}$ and $\alg{B}$, much more can be said about the clonoids between $\alg{A}$ and $\alg{B}$. Sparks showed in \cite{sparks-clonoids}, that there are only finitely many clonoids from $\alg{A}$ to $\alg{B}$ if $\alg{B}$ has a near-unanimity term, and at most countably many, if $\alg{B}$ has an edge-term. However, also a rich source algebra $\alg{A}$ can lead to finiteness results: As showed in \cite{LS-discriminator}, every clonoids from an algebra $\alg{A}$ with a discriminator term, to an algebra $\alg{B}$ on the same set, gives rise to only finitely many clonoids.

We would also like to highlight a series of papers by Lehtonen and coauthors \cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\alg{A},\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.

In this paper, we are interested in classifying clonoids, for which $\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \emph{difference clonoid}, see e.g. \cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\alg{A}$ and $\alg{B}$.

As an example of an application let us let us mention the classification of extensions of square-free groups in \cite{AM-Zpqextensions} and later \cite{fioravanti-groupexpansions}, which heavily relied on the classification of certain clonoids.

There is already a series of classification results in the literature. The clonoids between groups of prime order $\Z_p$ and $\Z_q$ were completely classified by Kreinecker \cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \neq q$ \cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).

In subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\alg{A}$ and $\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \cite{MW-clonoidsmodules}, and applies to all cases where $\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:

In recent years, generalizations of clones such as \emph{minions} and \emph{clonoids} have received increasing attention. Minions are sets of finitary operations from a set $A$ to a (possibly different) set $B$ that are closed under permutation and identification of variables, as well as the addition of dummy variables. Research on minions gained much traction due to their application on (promise) constraint satisfaction, see e.g. the survey \cite{BBKO-PCSP}. A \emph{clonoid} from an algebra $\alg{A}$ to an algebra $\alg{B}$ is a minion that is closed under (pre-)composition with operations of $\alg{A}$ and (post-)composition with the operations of $\alg{B}$. While the term `clonoid' was first introduced by Aichinger and Mayr in~\cite{AM-eqclasses}, the concept has appeared under various names earlier, e.g. in \cite{ CF-clonoids}.

Due to their weak structure, it is does not seem to be feasible to explicitly describe all minions between two given finite sets. A first hurdle is their sheer number: it was shown in \cite{pippenger-minions} that there are already continuum many minions between two 2-element sets. However, depending on the expressiveness of the algebras $\alg{A}$ and $\alg{B}$, much more can be said about the clonoids between $\alg{A}$ and $\alg{B}$. Sparks showed in \cite{sparks-clonoids}, that there are only finitely many clonoids from $\alg{A}$ to $\alg{B}$ if $\alg{B}$ has a near-unanimity term, and at most countably many, if $\alg{B}$ has an edge-term. However, also a rich source algebra $\alg{A}$ can lead to finiteness results: As showed in \cite{LS-discriminator}, every clonoids from an algebra $\alg{A}$ with a discriminator term, to an algebra $\alg{B}$ on the same set, gives rise to only finitely many clonoids.

We would also like to highlight a series of papers by Lehtonen and coauthors \cite{LN-minorsclique,LS-discriminator, CL-stability, lehtonen-majorityclonoids, lehtonen-NUclonoids}, which culminated in \cite{lehtonen-Booleanclonoids} to a complete characterization of all two-element algebras $(\alg{A},\alg{B})$, for which there are at most countable clonoids, together with a complete classification of clonoids in all such cases.

In this paper, we are interested in classifying clonoids, for which $\alg{B}$ is a module. Such clonoids often appear naturally when studying the term operations of Mal'cev algebras with a central congruence; this connection was recently made more explicit by Peter Mayr's definition of the \emph{difference clonoid}, see e.g. \cite{kompatscher-SMP2nil}. In particular, 2-nilpotent algebras give rise to clonoids between modules $\alg{A}$ and $\alg{B}$.

There is already a series of classification results in the literature. The clonoids between groups of prime order $\Z_p$ and $\Z_q$ were completely classified by Kreinecker \cite{kreinecker-zpclonoids} (in the case that $p=q$) and the first author, for $p \neq q$ \cite{fioravanti-clonoids}. Interestingly, in the latter case, there are only finitely many clonoids that are all generated by their unary functions (while for $p=q$ one obtains infinitely many).

In subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\alg{A}$ and $\alg{B}$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \cite{MW-clonoidsmodules}, and applies to all cases where $\alg{A}$ has a distributive lattice of submodules. This leads us to the following Conjecture:

\begin{abstract}
Clonoids are sets of finitary operations between two algebraic structures that are closed under composition with their term operations on both sides. We conjecture that, for finite modules $\algA$ and $\algB$ there are only finitely many clonoids from $\algA$ to $\algB$ if and only if $\algA$, $\algB$ are of coprime order.

We confirm this conjecture for a broad class of modules $\algA$. In particular we show that, if $\algA$ is a finite $k$-dimensional vector space, then every clonoid from $\alg A$ to a coprime module $\algB$ is generated by its $k$-ary functions (and arity $k-1$ does not suffice). In order to prove this results, we investigate `uniform generation by $(\algA,\algB)$-minors', a general criterion, which we show to apply to several other existing classifications results. Based on our analysis, we further prove that the subpower membership problem of certain 2-nilpotent Mal'cev algebras is solvable in polynomial time.
\end{abstract}

In subsequent generalizations of these results, it was also always observed that there are only finitely many clonoids between certain modules $\algA$ and $\algB$ of coprime order - the most recent, and most general such finiteness result is by Mayr and Wynne \cite{MW-clonoidsmodules}, and applies to all cases where $\algA$ has a distributive lattice of submodules. This leads us to the following Conjecture:

We remark that the ``only if direction'' follows from \cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.

The first main contribution of our paper is to confirm Conjecture \ref{conjecture:main} if $\algA$ is a finite vector space. In the case where $\algB$ is coprime to the vector space $\algA$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \ref{theorem:clonoidlattice}). In particular, this answers \cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\algA$ is the group $(\Z_p)^2$, for some prime $p$. By combining our result with \cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \ref{conjecture:main} for all modules $\algA =\F_1^{k_1}\times \cdots \times \F_n^{k_n} \times \alg D$ as $\F_1\times \cdots \times\F_n \times \algR$-module, where $\alg D$ is a distributive $\algR$-module, and all $\F_i$ are fields.

\begin{corollary} \label{corollary:main2}
Let $\F^k$ be the $k$-dimensional vector space over a finite field $\F$ and let $\algB$ be a finite module of coprime order. Then
\begin{enumerate}
\item there are only finitely many clonoids from $\F^k$ to $\algB$.
\item every clonoid from $\F^k$ to $\algB$ is finitely related by an at most $|F|^{k\times k}$-ary relation.
\end{enumerate}
\end{corollary}

\begin{corollary} \label{corollary:mainresultproducts}
Let $\algA =\F_1^{k_1}\times \cdots \times \F_n^{k_n} \times \alg D$ as $\F_1\times \cdots \times\F_n \times \algR$-module, where all $\F_i$ are finite fields, and $\alg D$ is a finite distributive $\algR$-module. Let $l$ be the nilpotence degree of the Jacobson radical of $\algR$, and let $k = \max(k_1,\ldots,k_n,l)$. Further let $\algB$ be an $\algS$-module, such that $|A|$ is invertible in $\algS$. Then
\begin{enumerate}
\item every clonoid from $\alg A$ to $\algB$ is generated by its $k$-ary part.
\item If $\algB$ is finite then there are only finitely many clonoids from $\algA$ to $\algB$.
\end{enumerate}
\end{corollary}

In this paper, we studied clonoids between finite modules. We proved that there are only finitely many clonoids from a finite vector space $\algA$ to a coprime module $\algB$, which is also in accordance to Conjecture \ref{conjecture:main}.

\begin{conjecture} \label{conjecture:main}
Let $\algA$ and $\algB$ be finite modules. Then, there are only finitely many clonoids from $\algA$ to $\algB$, if and only if $\algA$ and $\algB$ are of coprime order.
\end{conjecture}

\begin{theorem} \label{theorem:clonoidlattice}
The lattice of clonoids from $\F^k$ to $\algB$ is isomorphic to $\prod_{i=0}^k \Sub(\algM_{i,k}(\F,\algB))$.

Moreover, for $j\leq k$, the sublattice $\prod_{i=0}^j \Sub(\algM_{i,k}(\F,\algB))$ corresponds to the clonoids consisting of functions that are generated by their $j$-ary minors.
\end{theorem}

We remark that the ``only if direction'' follows from \cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.

 The first main contribution of our paper is to confirm Conjecture \ref{conjecture:main} if $\alg{A}$ is a finite vector space. In the case where $\alg{B}$ is coprime to the vector space $\alg{A}$, we are moreover able to give an explicit description of the lattice of all clonoids (Theorem \ref{theorem:clonoidlattice}). In particular, this answers \cite[Question 1.2]{MW-clonoidsmodules}, which asked for a classification of clonoids, if $\alg{A}$ is the group $(\Z_p)^2$, for some prime $p$. By combining our result with \cite{MW-clonoidsmodules}, we are further able to confirm Conjecture \ref{conjecture:main} for all modules $\alg{A} =\F_1^{k_1}\times \cdots \times \F_n^{k_n} \times \alg D$ as $\F_1\times \cdots \times\F_n \times \alg{R}$-module, where $\alg D$ is a distributive $\alg{R}$-module, and all $\F_i$ are fields.

Our classification is based on proving that, in the case of a vector space $\alg{A} = {\mathbf F}^k$ and coprime $\alg{B}$, the set of all operations from $A$ to $B$ is \emph{uniformly generated} by $k$-ary minors. While this technique was already hinted at in \cite{MW-clonoidsmodules}, we are going to discuss it in great detail in Section \ref{sec:unifgen}, and argue that many of the existing clonoid classifications follow the same pattern. We furthermore show that the arity $k$ is optimal, i.e., not every clonoid from ${\mathbf F}^k$ to $\alg{B}$ is generated by their $k-1$-ary functions. This coincides with a lower bound on the arity of generators of the \emph{full} clonoid between modules that we derive in Section \ref{sec:lowerbound}, and was already pointed out in the Bachelor thesis by Jan Van\v{e}\v{c}ek \cite{vanecek-thesis}.

Last, let us mention that, clonoids between affine algebras were also (sometimes implicitly) used to discuss some computational problems of Mal'cev algebras. The fact that some clonoids between coprime modules are generated by unary functions, which allows for a representation of their elements by nice normal forms, was used in \cite{KKK-CEQV2nil} to prove that checking polynomial identities in a given 2-nilpotent algebra can always be done in polynomial time. In \cite{mayr-VLloop} and \cite{KompatscherMayr2026} such normal forms were similarly used to find finite equational basis for some 2-nilpotent loops.

While we are not going to discuss such syntactic aspects of clonoids, we will apply our results to the \emph{subpower membership problem} of certain algebras. The subpower membership problem (SMP($\alg{A}$)), for a fixed finite algebra $\alg{A}$, is the computational problem whether a given partial operation $f\colon A^n \to A$ can be extended to a term operation of $\alg{A}$. In \cite{IMMVW-subpowers} it was asked, whether the subpower membership problem is always polynomial time solvable for algebras with few subpowers. This question, however remains unsolved even in the Mal'cev case. The easiest examples that are not covered by existing tractability results \cite{mayr-SMP, BMS-SMP}, are 2-nilpotent algebras, for which it was shown in \cite{kompatscher-SMP2nil} that the problem is polynomial time equivalent to a similar `interpolation problem' for their clonoids. In Section \ref{sect:SMP} we are going to use our results, to prove that the subpower membership problem of a big class of 2-nilpotent algebras is in P.

\subsection*{Organization of the paper}

The paper is organized as follows. In Section~\ref{sec:prel} we introduce the necessary concepts from universal algebra and fix some notation. In Section~\ref{sec:unifgen} we develop a theory of `uniform generation' and `uniform representation' by minors, a concept introduced in~\cite{MW-clonoidsmodules}. In particular we provide a purely combinatorial criterion for all $(\alg{A},\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem \ref{theorem:unigen}).

Section~\ref{sec:mainres} presents the main results of the paper, Theorem \ref{theorem:main} and discusses some consequences. In Section \ref{sect:SMP} we apply our results to the subpower membership problem. In Sections~\ref{sec:lowerbound} we provide a lower bound on the arity of the generators of the full clonoid between finite modules, which is sharp in our setting. Section~\ref{sec:conclusion} is dedicated to future work and possible applications of our results.

We remark that the ``only if direction'' follows from \cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.

For confirming Conjecture~\ref{conjecture:main} when $\alg{A}$ is a finite vector space and $\alg{B}$ is coprime to $\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\alg{A}={\mathbf F}^k$ and coprime $\alg{B}$, “the set of all operations from $A$ to $B$ is \emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \cite{MW-clonoidsmodules},” and they “discuss it in great detail in Section~\ref{sec:unifgen},” including “a purely combinatorial criterion for all $(\alg{A},\alg{B})$-clonoids to be generated by their $k$-ary functions (Theorem~\ref{theorem:unigen}).” They also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\mathbf F}^k$ to $\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \emph{full} clonoid between modules” derived in Section~\ref{sec:lowerbound}. Finally, they state that “by combining our result with \cite{MW-clonoidsmodules}, we are further able to confirm Conjecture~\ref{conjecture:main}” for broader classes of modules $\alg{A}=\F_1^{k_1}\times\cdots\times\F_n^{k_n}\times\alg D.

We remark that the ``only if direction'' follows from \cite[Theorem $1.3$]{MW-clonoidsmodules}, where the authors proved that for modules with a common divisor, one can always construct an infinite ascending chain of clonoids.

For confirming the main conjecture when $\alg{A}$ is a finite vector space and $\alg{B}$ is coprime to $\alg{A}$, the paper’s approach is described as follows: the classification is based on proving that, for a vector space $\alg{A}={\mathbf F}^k$ and coprime $\alg{B}$, “the set of all operations from $A$ to $B$ is \emph{uniformly generated} by $k$-ary minors.” The authors note this technique was “already hinted at in \cite{MW-clonoidsmodules},” and they “discuss it in great detail” later, including “a purely combinatorial criterion for all $(\alg{A},\alg{B})$-clonoids to be generated by their $k$-ary functions.” We first recall this criterion.

\begin{theorem} \label{theorem:unigen}
Let $\cloA$, $\cloB$ be clones, $n \geq 1$, and let $\cloA_{const} = \Clo(\cloA \cup \{a\}_{a\in A})$ be the clone generated by $\cloA$ and all constant operations over its set. Let $\clodC$ a clonoid from $\cloA_{const}$
to $\cloB$. Then the following are equivalent:
\begin{enumerate}
\item\label{ite:th_unigen1} $\clodC^{(n+1)}$ is $(n,\cloA,\cloB)$-UG
\item\label{ite:th_unigen2} $\clodC^{(k)}$ is $(n,\cloA,\cloB)$-UG for some $k>n$
\item\label{ite:th_unigen3} $\clodC$ is $(n,\cloA,\cloB)$-UG
\item\label{ite:th_unigen4} $\forall k,l \in \N$, every partial operation $I \colon \clodC^{(k)} \to \clodC^{(l)}$
that is $(\cloA,\cloB)$-UR is $(n,\cloA,\cloB)$-UR.
\end{enumerate}
\end{theorem}

They also establish an optimality statement for this method: “the arity $k$ is optimal, i.e., not every clonoid from ${\mathbf F}^k$ to $\alg{B}$ is generated by their $k-1$-ary functions,” and relate this to a “lower bound on the arity of generators of the \emph{full} clonoid between modules” derived later. Finally, they state that “by combining our result with \cite{MW-clonoidsmodules}, we are further able to confirm the main conjecture” for broader classes of modules $\alg{A}=\F_1^{k_1}\times\cdots\times\F_n^{k_n}\times\alg D.

Question: Let \(\alg A\) and \(\alg B\) be finite modules. A clonoid from \(\alg A\) to \(\alg B\) is a set of finitary functions from powers of \(A\) to \(B\) that is closed under taking minors (permuting variables, identifying variables, and adding dummy variables), as well as under precomposition with term operations of \(\alg A\) and postcomposition with term operations of \(\alg B\). Two finite modules are said to be of coprime order when \(\gcd(|A|,|B|)=1\). Which statement holds about the number of clonoids from \(\alg A\) to \(\alg B\)?

There are only finitely many clonoids from \(\alg A\) to \(\alg B\) if and only if \(\alg A\) and \(\alg B\) are of coprime order.

Choices:
B. There are only finitely many clonoids from \(\alg A\) to \(\alg B\) whenever \(\alg A\) and \(\alg B\) have distinct orders.
C. If \(\alg A\) and \(\alg B\) are of coprime order, then there are only finitely many clonoids from \(\alg A\) to \(\alg B\).
D. There are only finitely many clonoids from \(\alg A\) to \(\alg B\) if and only if every clonoid from \(\alg A\) to \(\alg B\) is generated by its unary functions.
E. There are only finitely many clonoids from \(\alg A\) to \(\alg B\) if and only if, for some integer \(n\ge 1\) depending only on \(|A|\), every clonoid from \(\alg A\) to \(\alg B\) is generated by its \(n\)-ary functions.

coprime-order hypothesis replaced by distinct-order condition

dropped the converse direction from the iff statement

optimal k-ary generation confused with unary generation

uniform generation bound made to depend only on |A| and asserted equivalent to finiteness for all modules


Question: Let \(\alg A\) and \(\alg B\) be finite modules. A clonoid from \(\alg A\) to \(\alg B\) is a set of finitary operations \(A^n\to B\) that is closed under precomposition with term operations of \(\alg A\) and postcomposition with term operations of \(\alg B\). Which statement is equivalent to saying that there are only finitely many clonoids from \(\alg A\) to \(\alg B\)?

\(\alg A\) and \(\alg B\) have coprime orders; equivalently, \(\gcd(|A|,|B|)=1\).

Choices:
B. \(\alg A\) and \(\alg B\) have relatively prime composition lengths; equivalently, no prime occurs with positive multiplicity in both Jordan--H\"older lengths.
C. \(\alg A\) and \(\alg B\) do not have the same cardinality.
D. For every finite module \(\alg B\), if there are only finitely many clonoids from \(\alg A\) to \(\alg B\), then \(\gcd(|A|,|B|)=1\).
E. \(\alg A\) and \(\alg B\) have coprime orders, and moreover every clonoid from \(\alg A\) to \(\alg B\) is generated by its unary functions.

replaced coprime-order condition by the weaker consequence of unequal sizes

iff weakened to only-if implication with quantifier shift over target module

optimal k-ary generation for vector spaces strengthened to unary generation for all finite modules


\begin{theorem}\label{main theorem k+1-good}
Let $\rho$ be a Schwartz function on $\R^{k+1}$ with $\int \rho = 1$. Assume that
\[
	\supp \hat \rho \subset (-\inj M, \inj M)^{k+1}.
\]
Let $\Gamma$ be a $k$-good cone. Then for $\tau = (\tau_1, \tau_2, \dots, \tau_{k+1}) \in \Gamma$,
\[
	\rho * \mu(\tau) = (2\pi)^{-kn} \vol M \vol F^{-1}(\tau) + O(|\tau|^{k(n-1)-2}),
\]
where the constants implicit in the big-$O$ notation depend on $M$, $\Gamma$, and $\rho$ but not on $\tau$.
\end{theorem}

\begin{theorem}\label{main theorem k+1-gon-bad}
	For each $\epsilon>0$ and integer $N > 0$, there exists a constant $C_{\epsilon, N}$ such that for any positive numbers $\tau_{1},\tau_{2},\ldots,\tau_{k}$, we have
	\[
		\sum_{\lambda_{i_{k+1}}\geq (1+\epsilon)(\tau_1+\tau_2+\dots+\tau_k)} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2 \leq C_{\epsilon, N}(\tau_1+\tau_2+\dots+\tau_k)^{-N},
	\]
	where $\lambda_{i_j} \leq \tau_j$ for $1 \leq j \leq k$.
\end{theorem}

\begin{figure}
\includegraphics[width=0.6\textwidth]{2}
\caption{A pentagon with side lengths $a_1, \ldots, a_5$.}
\label{2}
\includegraphics[width=0.6\textwidth]{3}
\caption{A self-intersecting pentagon with the same side lengths.}
\label{3}
\end{figure}

\begin{proposition}
 [Scaling of $F$]\label{Scaling of F} Let $\Gamma$ be a $k$-good cone and fix $\tau \in \Gamma$. Then $F^{-1}(\tau) \neq \emptyset$ and $\tau$ is a regular value of $F$, so $F^{-1}(\tau)$ is a smooth compact submanifold of codimension $k+1$. Let
\[
d:=k n-(k+1)=k(n-1)-1
\]
be its dimension. Then $0<\operatorname{vol} F^{-1}(\tau)<\infty$ and, for every $r>0$,
$$
\operatorname{vol} F^{-1}(r \tau)=r^d \operatorname{vol} F^{-1}(\tau).
$$
\end{proposition}

\subsection{Background}

Let $M$ be a compact Riemannian manifold without boundary, and $e_1, e_2, \ldots$ be an orthonormal basis of Laplace-Beltrami eigenfunctions with
\[
	\Delta_g e_j = -\lambda_j^2 e_j,
\]
where $\lambda_j$ is the frequency of $e_j$.

We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion
\begin{equation}\label{harmonic expansion}
	e_{i_1} e_{i_2} \dots e_{i_k} = \sum_{i_{k+1}} \langle e_{i_1} e_{i_2} \dots e_{i_k}, e_{i_{k+1}} \rangle e_{i_{k+1}}.
\end{equation}
The classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \dots e_{i_k}$. If we set $\lambda_{i_1}=\lambda_{i_2}=\dots =\lambda_{i_k}=\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\lambda_j}$. For this problem, Sogge provided a complete answer for $k\geq 1$, see \cite{MR930395}.
However, for general eigenvalues $\lambda_{i_1}, \lambda_{i_2}, \ldots, \lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \eqref{harmonic expansion} as follows:
\begin{equation}\label{rewrite harmonic expansion}
	\|e_{i_1} e_{i_2} \cdots e_{i_k}\|_{L^2}^2 = \sum_{i_{k+1}\in\mathbb Z^+} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2.
\end{equation}
The decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}}$ do the coefficients $\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle$ exhibit decay, and how is their $\ell^2$-mass distributed as these frequencies vary?

When $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\langle e_j^2, e_\ell \rangle$ decay exponentially in $\lambda_\ell$ \cite{MR1277052}. Bernstein, Reznikov, Kr\"otz, and Stanton obtained the optimal estimate \cite{MR1715328, MR2081437}. For the more general coefficients $\langle e_i e_j, e_\ell \rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \cite{MR3997636}. More recently, the first author improved their estimate \cite{MR4376455}. Given $\epsilon > 0$ and an integer $N \geq 1$, there exists a constant $C_{\epsilon,N}$ for which
\begin{equation}\label{Emmett conclusion}
	\sum_{\lambda_\ell \geq (2+\epsilon)\lambda} |\langle e_i e_j, e_\ell \rangle|^2 \leq C_{\epsilon, N} \lambda^{-N} \qquad \text{for all $i$, $j$ with $\lambda_i, \lambda_j \leq \lambda$.}
\end{equation}

Furthermore, he proved that the $(2+\epsilon)\lambda$ factor is nearly optimal. He also obtained an asymptotic expansion for the non-rapidly decaying part \cite{MR4376455}. Building on this work, we study the multi-product of $(k+1)$ eigenfunctions and obtain analogous results.
One of the most striking features of \cite{MR4376455} is that it connects triple products of eigenfunctions to geometric configurations, namely triangles. We extend this perspective to general $k \geq 2$, where the corresponding configurations are naturally described by polygons.

\subsection{Polygons}

Based on \eqref{rewrite harmonic expansion}, we study the sums
\begin{equation}\label{single sums}
	\sum_{(\lambda_{i_1}, \lambda_{i_2}, \dots ,\lambda_{i_{k+1}}) = (a_1, a_2, \dots ,a_{k+1})} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2
\end{equation}
for given eigenvalues $a_1, a_2, \dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.

The main point of the present paper is to show that the sum \eqref{single sums} can be estimated by counting configurations of $(k+1)$-gons with side lengths $a_1, a_2, \dots, a_{k+1}$. To state our results, we first introduce the following definition concerning the side lengths of a $(k+1)$-gon.

\begin{definition}\label{k+1-good and k+1-gon-bad}
A vector $\tau = (\tau_1, \ldots, \tau_{k+1}) \in (0,\infty)^{k+1}$ is called $k$-good if it satisfies the following two conditions.
\begin{enumerate}
\item the $(k+1)$-gon inequality: for every $1 \leq j \leq k+1$,
\begin{equation}\label{2.1}
\tau_j < \sum_{\ell \neq j} \tau_{\ell}.
\end{equation}
\item the nondegeneracy condition: for every sign vector $\varepsilon = (\varepsilon_1, \ldots, \varepsilon_{k+1}) \in \{ \pm 1\}^{k+1}$,
\begin{equation}\label{2.2}
|\varepsilon \cdot \tau| = \left|\sum_{j=1}^{k+1} \varepsilon_j \tau_j\right| > 0.
\end{equation}
\end{enumerate}
We denote the set of $k$-good vectors by $\Gamma_g$.

A vector $\tau = (\tau_1, \ldots, \tau_{k+1}) \in (0,\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \leq j \leq k+1$, such that
\begin{equation}\label{2.3}
\tau_j > \sum_{\ell \neq j} \tau_{\ell}.
\end{equation}
We denote the set of $k$-bad vectors by $\Gamma_b$.

A closed cone $\Gamma \subset (0,\infty)^{k+1}$ is called $k$-good if $\Gamma \subset \Gamma_g$, and it is called $k$-bad if $\Gamma \subset \Gamma_b$.
\end{definition}

The set of $\tau$ satisfying condition (1) consists of side-length data that can be realized by a $(k+1)$-gon in the plane, whereas the set of $k$-bad vectors contains no such points. The set of $k$-good vectors is obtained from the set of $\tau$ satisfying condition (1) by removing those $\tau$ for which a linear combination $|\varepsilon \cdot \tau|$ vanishes for some $\varepsilon \in \{ \pm 1\}^{k+1}$. For example, when $k=3$, the vector $(2,2,2,2)$ satisfies the $(k+1)$-gon inequality, but $(2,2,2,2)\notin \Gamma_g$ since it fails the nondegeneracy condition. Concretely, consider the degenerate quadrilateral in the plane formed by the vectors $(2,0), (-2,0), (2,0), (-2,0)$. Each side has length $2$, but the polygon is collinear and has zero area. In this case, $\tau \notin \Gamma_g$, as illustrated in Figure \ref{4}. When $k=2$, the nondegeneracy condition is automatic once one assumes condition (1), due to the rigidity of triangles. This is the primary reason it was not clear whether triple-product results could be generalized to higher multi-products, where such rigidity is absent. Our nondegeneracy condition is designed to fill this gap. As we will see in the proof, it ensures that several crucial steps go through and yields clean asymptotics.

\begin{figure}
\includegraphics[width=0.6\textwidth]{4}
\caption{An example of $\tau\notin\Gamma_g$.}
\label{4}
\end{figure}
In what follows, we establish a connection between \eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\chi_\lambda$ and the theory of Fourier integral operators. Our argument also shows that \eqref{single sums} decays rapidly in the regime where $(\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.

\subsection{Main results}
Our first theorem shows that the rapidly decaying contribution to \eqref{single sums} comes from frequencies $(\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.

We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion
\begin{equation}\label{harmonic expansion}
 e_{i_1} e_{i_2} \dots e_{i_k} = \sum_{i_{k+1}} \langle e_{i_1} e_{i_2} \dots e_{i_k}, e_{i_{k+1}} \rangle e_{i_{k+1}}.
\end{equation}
The classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \dots e_{i_k}$. If we set $\lambda_{i_1}=\lambda_{i_2}=\dots =\lambda_{i_k}=\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\lambda_j}$. For this problem, Sogge provided a complete answer for $k\geq 1$, see \cite{MR930395}.
However, for general eigenvalues $\lambda_{i_1}, \lambda_{i_2}, \ldots, \lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \eqref{harmonic expansion} as follows:
\begin{equation}\label{rewrite harmonic expansion}
 \|e_{i_1} e_{i_2} \cdots e_{i_k}\|_{L^2}^2 = \sum_{i_{k+1}\in\mathbb Z^+} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2.
\end{equation}
The decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}}$ do the coefficients $\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle$ exhibit decay, and how is their $\ell^2$-mass distributed as these frequencies vary?

When $k=2$, as in many spectral problems, this question is closely connected to number theory in the automorphic setting. Sarnak showed that for fixed $j$ the coefficients $\langle e_j^2, e_\ell \rangle$ decay exponentially in $\lambda_\ell$ \cite{MR1277052}. Bernstein, Reznikov, Kr\"otz, and Stanton obtained the optimal estimate \cite{MR1715328, MR2081437}. For the more general coefficients $\langle e_i e_j, e_\ell \rangle$, a classical decay estimate in terms of the eigenvalues was given by Lu, Sogge, and Steinerberger \cite{MR3997636}. More recently, the first author improved their estimate \cite{MR4376455}. Given $\epsilon > 0$ and an integer $N \geq 1$, there exists a constant $C_{\epsilon,N}$ for which
\begin{equation}\label{Emmett conclusion}
 \sum_{\lambda_\ell \geq (2+\epsilon)\lambda} |\langle e_i e_j, e_\ell \rangle|^2 \leq C_{\epsilon, N} \lambda^{-N} \qquad \text{for all $i$, $j$ with $\lambda_i, \lambda_j \leq \lambda$.}
\end{equation}

Based on \eqref{rewrite harmonic expansion}, we study the sums
\begin{equation}\label{single sums}
 \sum_{(\lambda_{i_1}, \lambda_{i_2}, \dots ,\lambda_{i_{k+1}}) = (a_1, a_2, \dots ,a_{k+1})} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2
\end{equation}
for given eigenvalues $a_1, a_2, \dots, a_{k+1}$. These sums are independent of the choice of eigenbasis; hence their behavior is determined solely by the underlying manifold and its metric.

A vector $\tau = (\tau_1, \ldots, \tau_{k+1}) \in (0,\infty)^{k+1}$ is called $k$-bad if there exists $j$, $1 \leq j \leq k+1$, such that
\begin{equation}\label{2.3}
\tau_j > \sum_{\ell \neq j} \tau_{\ell}.
\end{equation}
We denote the set of $k$-bad vectors by $\Gamma_b$.

\begin{figure}
\includegraphics[width=0.6\textwidth]{4}
\caption{An example of $\tau\notin\Gamma_g$.}
\label{4}
\end{figure}
In what follows, we establish a connection between \eqref{single sums} and the count of configurations of $(k+1)$-gons using the spectral projection operator $\chi_\lambda$ and the theory of Fourier integral operators. Our argument also shows that \eqref{single sums} decays rapidly in the regime where $(\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}})$ fails to satisfy the $(k+1)$-gon inequality.

\subsection{Main results}
Our first theorem shows that the rapidly decaying contribution to \eqref{single sums} comes from frequencies $(\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.

\begin{equation}\label{harmonic expansion}
	e_{i_1} e_{i_2} \dots e_{i_k} = \sum_{i_{k+1}} \langle e_{i_1} e_{i_2} \dots e_{i_k}, e_{i_{k+1}} \rangle e_{i_{k+1}}.
\end{equation}

\begin{equation}\label{rewrite harmonic expansion}
	\|e_{i_1} e_{i_2} \cdots e_{i_k}\|_{L^2}^2 = \sum_{i_{k+1}\in\mathbb Z^+} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2.
\end{equation}

\begin{equation}\label{single sums}
	\sum_{(\lambda_{i_1}, \lambda_{i_2}, \dots ,\lambda_{i_{k+1}}) = (a_1, a_2, \dots ,a_{k+1})} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2
\end{equation}

We are interested in the general behavior of the multi-product of $k$ eigenfunctions $e_{i_1} e_{i_2} \dots e_{i_k}$. To analyze this multi-product more conveniently, we express it as a sum of eigenfunctions with coefficients by a harmonic expansion
\begin{equation}\label{harmonic expansion}
 e_{i_1} e_{i_2} \dots e_{i_k} = \sum_{i_{k+1}} \langle e_{i_1} e_{i_2} \dots e_{i_k}, e_{i_{k+1}} \rangle e_{i_{k+1}}.
\end{equation}
The classical question is to study the decay of the $L^2$ norm of $e_{i_1} e_{i_2} \dots e_{i_k}$. If we set $\lambda_{i_1}=\lambda_{i_2}=\dots =\lambda_{i_k}=\lambda_j$, then the problem reduces to study the $L^{2k}$ norm of the eigenfunction $e_{\lambda_j}$. For this problem, Sogge provided a complete answer for $k\geq 1$, see \cite{MR930395}.
However, for general eigenvalues $\lambda_{i_1}, \lambda_{i_2}, \ldots, \lambda_{i_k}$, the situation is more complicated. By orthogonality, we can write the $L^2$ norm of \eqref{harmonic expansion} as follows:
\begin{equation}\label{rewrite harmonic expansion}
 \|e_{i_1} e_{i_2} \cdots e_{i_k}\|_{L^2}^2 = \sum_{i_{k+1}\in\mathbb Z^+} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2.
\end{equation}
The decay of the terms on the right-hand side therefore determines the behavior of the left-hand side. We thus aim to study both the decay and how this decay is distributed in \eqref{rewrite harmonic expansion}. This leads to the following concrete problem: under what conditions on the frequencies $\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}}$ do the coefficients $\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle$ exhibit decay, and how is their $\ell^2$-mass distributed as these frequencies vary?

\subsection{Main results}
Our first theorem shows that the rapidly decaying contribution to \eqref{single sums} comes from frequencies $(\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}})$ whose coordinates cannot occur as the side lengths of a $(k+1)$-gon.

The hypothesis $\lambda_{i_{k+1}} \geq (1+\epsilon)(\tau_1+\cdots+\tau_k)$ forces the $(k+1)$-tuple $(\tau_1,\ldots,\tau_k,\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\lambda_{i_1},\ldots,\lambda_{i_k},\lambda_{i_{k+1}})$ with $\lambda_{i_j} \leq \tau_j$ for $1 \leq j \leq k$ is also $k$-bad.

For clarity, we do not organize the left-hand side by summing over all tuples $(\lambda_{i_1},\ldots,\lambda_{i_{k+1}})$ lying in a specified region of $(0,\infty)^{k+1}$. Instead, we fix upper bounds $\tau_j$ for $\lambda_{i_j}$, $1 \leq j \leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\lambda_{i_{k+1}} \geq (1+\epsilon)(\tau_1+\cdots+\tau_k)$. The proof uses the method of non-stationary phase.
We record a flat-torus example in Section \ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \ref{main theorem k+1-gon-bad} is essentially sharp.

Our second and main theorem concerns the contribution from $k$-good vectors $\tau \in (0,\infty)^{k+1}$.
 Following \cite{MR4376455}, we introduce the joint spectral measure
\begin{equation}\label{joint spectral measure}
 \mu = \sum_{i_1,i_2,\dots,i_{k+1}} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2 \, \delta_{(\lambda_{i_1},\lambda_{i_2},\dots,\lambda_{i_{k+1}})}
\end{equation}
on $\R^{k+1}$, where $\delta$ denotes the Dirac measure. Thus $\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle$. Our second theorem describes the mass of $\mu$ carried by frequencies $(\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}})$ lying in $k$-good cones.

Next we verify that $\tau$ is a regular value of $F$. Fix $\xi=(\xi_1,\ldots,\xi_k)\in F^{-1}(\tau)$ and set
\[
\omega_j=\frac{\xi_j}{|\xi_j|}\in S^{n-1}\ (1\le j\le k),\qquad
\omega_{k+1}=\frac{\xi_1+\cdots+\xi_k}{|\xi_1+\cdots+\xi_k|}\in S^{n-1}.
\]
For $h=(h_1,\ldots,h_k)\in\R^{kn}$ we compute
\[
d(|\xi_j|)[h]=\langle \omega_j,h_j\rangle,\qquad
d(|\xi_1+\cdots+\xi_k|)[h]=\Bigl\langle \omega_{k+1},\sum_{j=1}^k h_j\Bigr\rangle.
\]
If $\mathrm{rank}(dF_\xi)<k+1$, these $k+1$ linear functionals are linearly dependent, so there exist
$(c_1,\ldots,c_k,c_{k+1})\neq 0$ such that for all $h$,
\[
\sum_{j=1}^k c_j\langle \omega_j,h_j\rangle
+c_{k+1}\Bigl\langle \omega_{k+1},\sum_{j=1}^k h_j\Bigr\rangle=0.
\]
Equivalently,
\[
\sum_{j=1}^k \langle c_j\omega_j+c_{k+1}\omega_{k+1},\ h_j\rangle=0
\quad\text{for all }h_1,\ldots,h_k,
\]
hence $c_j\omega_j+c_{k+1}\omega_{k+1}=0$ for every $j$. In particular $c_{k+1}\neq 0$, and
$\omega_j=\varepsilon_j\,\omega_{k+1}$ for some $\varepsilon_j\in\{\pm1\}$. Therefore
\[
\xi_1+\cdots+\xi_k=\sum_{j=1}^k |\xi_j|\omega_j
=\Bigl(\sum_{j=1}^k \varepsilon_j|\xi_j|\Bigr)\omega_{k+1},
\]
and taking norms gives
\[
\tau_{k+1}=|\xi_1+\cdots+\xi_k|
=\Bigl|\sum_{j=1}^k \varepsilon_j|\xi_j|\Bigr|
=\Bigl|\sum_{j=1}^k \varepsilon_j\,\tau_j\Bigr|.
\]
Let $s:=\sum_{j=1}^k \varepsilon_j\,\tau_j$ and choose $\varepsilon_{k+1}\in\{\pm1\}$ so that
$\varepsilon_{k+1}\tau_{k+1}+s=0$. Then the sign vector
\[
\varepsilon=(\varepsilon_1,\ldots,\varepsilon_k,\varepsilon_{k+1})\in\{\pm1\}^{k+1}
\]
satisfies $\varepsilon\cdot\tau=0$, contradicting the nondegeneracy condition \eqref{2.2}.
Hence $\mathrm{rank}(dF_\xi)=k+1$ for all $\xi\in F^{-1}(\tau)$, so $\tau$ is a regular value.

\begin{lemma}\label{lem:kplus1-eigenfunction-product}
Let $\epsilon>0$ and set
\[
T_m := (1+\epsilon)(\tau_1+\cdots+\tau_k)+m ,\qquad m\in\N.
\]
Assume that $\lambda_{i_j}\le \tau_j$ for $1\le j\le k$ and that
\[
\lambda_{i_{k+1}}\in [T_m,T_m+1].
\]
Then for every integer $N\ge 0$ there exists a constant $C_{\epsilon,N}$ such that
\[
\bigl|\langle e_{i_1} e_{i_2}\cdots e_{i_k},\, e_{i_{k+1}}\rangle\bigr|
\le C_{\epsilon,N}\, T_m^{-N}.
\]
\end{lemma}

\begin{proof}[Proof of Theorem \ref{main theorem k+1-gon-bad}]
Decompose the tail into unit spectral windows:
\begin{equation*}
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\cdots+\tau_k)}
\bigl|\langle e_{i_1}\cdots e_{i_k}, e_{i_{k+1}}\rangle\bigr|^2
=\sum_{m=0}^\infty \sum_{\lambda_{i_{k+1}}\in[T_m,T_m+1]}
\bigl|\langle e_{i_1}\cdots e_{i_k}, e_{i_{k+1}}\rangle\bigr|^2
\end{equation*}
By the Weyl law,
\[
\#\{\lambda_{i_{k+1}}\in[T_m,T_m+1]\}\ \lesssim\ T_m^{n-1}.
\]
Lemma \ref{lem:kplus1-eigenfunction-product} gives
\[
\sup_{\lambda_{i_{k+1}}\in[T_m,T_m+1]}
\bigl|\langle e_{i_1}\cdots e_{i_k}, e_{i_{k+1}}\rangle\bigr|^2
\ \lesssim_{\epsilon,N}\ T_m^{-2N}.
\]
Hence
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\cdots+\tau_k)}
\bigl|\langle e_{i_1}\cdots e_{i_k}, e_{i_{k+1}}\rangle\bigr|^2
\ \lesssim_{\epsilon,N}\ \sum_{m=0}^\infty T_m^{\,n-1-2N}.
\]
Given $N'>0$, choose $N$ so that $2N-(n-1) > N'$. Then the last sum is bounded by
$C_{\epsilon,N'}(\tau_1+\cdots+\tau_k)^{-N'}$, which proves the theorem.
\end{proof}

The hypothesis $\lambda_{i_{k+1}} \geq (1+\epsilon)(\tau_1+\cdots+\tau_k)$ forces the $(k+1)$-tuple $(\tau_1,\ldots,\tau_k,\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others. In particular, any $(\lambda_{i_1},\ldots,\lambda_{i_k},\lambda_{i_{k+1}})$ with $\lambda_{i_j} \leq \tau_j$ for $1 \leq j \leq k$ is also $k$-bad.

For clarity, we do not organize the left-hand side by summing over all tuples $(\lambda_{i_1},\ldots,\lambda_{i_{k+1}})$ lying in a specified region of $(0,\infty)^{k+1}$. Instead, we fix upper bounds $\tau_j$ for $\lambda_{i_j}$, $1 \leq j \leq k$, and then estimate the tail sum over those $i_{k+1}$ with $\lambda_{i_{k+1}} \geq (1+\epsilon)(\tau_1+\cdots+\tau_k)$. The proof uses the method of non-stationary phase.
We record a flat-torus example in Section \ref{sec:flat-torus-example} showing that the frequency threshold in Theorem \ref{main theorem k+1-gon-bad} is essentially sharp.

Our second and main theorem concerns the contribution from $k$-good vectors $\tau \in (0,\infty)^{k+1}$.
 Following \cite{MR4376455}, we introduce the joint spectral measure
\begin{equation}\label{joint spectral measure}
	\mu = \sum_{i_1,i_2,\dots,i_{k+1}} \bigl|\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle\bigr|^2 \, \delta_{(\lambda_{i_1},\lambda_{i_2},\dots,\lambda_{i_{k+1}})}
\end{equation}
on ${\mathbb R}^{k+1}$, where $\delta$ denotes the Dirac measure. Thus $\mu$ is a weighted sum of point masses, with weights given by the squared magnitudes of the coefficients $\langle e_{i_1} e_{i_2} \cdots e_{i_k}, e_{i_{k+1}} \rangle$. Our second theorem describes the mass of $\mu$ carried by frequencies $(\lambda_{i_1}, \lambda_{i_2}, \dots, \lambda_{i_{k+1}})$ lying in $k$-good cones.

Before stating the result, we introduce some terminology by recalling the definition and basic properties of the Leray density. Let $F : {\mathbb R}^n \to {\mathbb R}^d$ be a smooth function, and let $y_0 \in {\mathbb R}^d$ be such that the differential $dF$ has full rank at every point of the level set $F^{-1}(y_0)$. By the inverse function theorem, $F^{-1}(y_0)$ is a smooth $(n-d)$-dimensional submanifold. Choose local coordinates $(z,z')$ in a neighborhood of a point of $F^{-1}(y_0)$ so that $z$ parametrizes $F^{-1}(y_0)$ and $\det(\partial F/\partial z') \neq 0$. The Leray density on $F^{-1}(y_0)$ is defined by
\begin{equation}\label{Leray density definition}
	d_F = \left|\det \frac{\partial F}{\partial z'}\right|^{-1} \, dz,
\end{equation}
and is independent of the choice of complementary coordinates $z'$. The corresponding Leray volume is
\[
	\operatorname{vol} F^{-1}(y_0) = \int_{F^{-1}(y_0)} d_F.
\]

Given a continuous function $f$ defined on a neighborhood of $F^{-1}(y_0)$, the integral of $f$ over $F^{-1}(y_0)$ with respect to the Leray measure can be expressed using the Dirac distribution as
\[
	\int_{F^{-1}(y_0)} f \, d_F = \int_{{\mathbb R}^n} f(x) \delta(F(x) - y_0) \, dx.
\]

For the purposes of this paper, we set $F : {\mathbb R}^{kn} \to {\mathbb R}^{k+1}$ by
\begin{equation}\label{def F}
	F(\xi_1,\xi_2,\dots,\xi_k) = \bigl(|\xi_1|,|\xi_2|,\dots,|\xi_k|,\ |\xi_1 + \cdots + \xi_k|\bigr).
\end{equation}
Then the level set $F^{-1}(\tau)$ consists of $k$-tuples $(\xi_1,\xi_2,\dots,\xi_k)$ satisfying
\[
|\xi_j|=\tau_j \ \ (1\le j\le k)
\qquad \text{and} \qquad
|\xi_1+\cdots+\xi_k|=\tau_{k+1}.
\]
Such a $k$-tuple determines a closed $(k+1)$-gon in ${\mathbb R}^n$ with side lengths $\tau_1,\tau_2,\dots,\tau_{k+1}$, where the final side is the closing vector $-(\xi_1+\cdots+\xi_k)$.
The next theorem provides a description of the concentration of $\mu$ in $k$-good cones.

\begin{theorem}\label{main theorem k+1-good}
Let $\rho$ be a Schwartz function on ${\mathbb R}^{k+1}$ with $\int \rho = 1$. Assume that
\[
	\operatorname{supp} \hat \rho \subset (-\operatorname{inj} M, \operatorname{inj} M)^{k+1}.
\]
Let $\Gamma$ be a $k$-good cone. Then for $\tau = (\tau_1, \tau_2, \dots, \tau_{k+1}) \in \Gamma$,
\[
	\rho * \mu(\tau) = (2\pi)^{-kn} \operatorname{vol} M \operatorname{vol} F^{-1}(\tau) + O(|\tau|^{k(n-1)-2}),
\]
where the constants implicit in the big-$O$ notation depend on $M$, $\Gamma$, and $\rho$ but not on $\tau$.
\end{theorem}

Since $\rho$ is rapidly decaying and $\operatorname{supp} \hat \rho$ is contained in $(-\operatorname{inj} M, \operatorname{inj} M)^{k+1}$, the convolution $\rho * \mu$ provides a smoothed version of $\mu$ that captures the mass of $\mu$ near $\tau$. In particular, Theorem \ref{main theorem k+1-good} gives an asymptotic description of this smoothed local mass for $\tau \in \Gamma$. Next, we determine the growth rate of $\operatorname{vol} F^{-1}(\tau)$ as $|\tau| \to \infty$. When $k=2$, this quantity is related to the area of the corresponding triangle \cite{MR4376455}. For $k>2$, the shape of a $(k+1)$-gon with fixed side lengths is not uniquely determined (Figures \ref{2} and \ref{3}), so it is not sufficient to consider only its planar area. We prove in Proposition \ref{Scaling of F} that, under our assumptions, $\operatorname{vol} F^{-1}(\tau)$ is of order $|\tau|^{k(n-1)-1}$, so the remainder term $O(|\tau|^{k(n-1)-2})$ is indeed lower order. As in \cite{MR4376455}, the leading term can be interpreted in terms of the volume of the configuration space of $(k+1)$-gons with side lengths prescribed by $\tau$.
\subsection*{Organization}
In Section~2 we analyze the configuration space that appears in the main term of Theorem~\ref{main theorem k+1-good}, and in particular prove the scaling law for the relevant fiber volume. Section~3 contains the proof of Theorem~\ref{main theorem k+1-gon-bad}, establishing rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument. Section~4 proves Theorem~\ref{main theorem k+1-good} by expressing the Fourier transform of the joint spectral measure in terms of Fourier integral operators and computing the associated symbolic data using clean composition and half-density calculus. Finally, Section~\ref{sec:flat-torus-example} presents a flat-torus example showing that the frequency cutoff in Theorem~\ref{main theorem k+1-gon-bad} is essentially sharp.

\subsection*{Acknowledgement} This project is supported by the National Key R\&D Program of China under Grant No. 2022YFA1007200, the Natural Science Foundation of China under Grant No. 12571107, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR25A010001.

The introduction explains that the condition $\lambda_{i_{k+1}} \ge (1+\epsilon)(\tau_1+\cdots+\tau_k)$ “forces the $(k+1)$-tuple $(\tau_1,\ldots,\tau_k,\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\lambda_{i_1},\ldots,\lambda_{i_k},\lambda_{i_{k+1}})$ with $\lambda_{i_j}\le \tau_j$ is also $k$-bad.

Rather than summing over a region in $(0,\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\tau_j$ for $\lambda_{i_j}$, $1\le j\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\lambda_{i_{k+1}} \ge (1+\epsilon)(\tau_1+\cdots+\tau_k)$.” The proof of Theorem~\ref{main theorem k+1-gon-bad} then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”

The introduction explains that the condition $\lambda_{i_{k+1}} \ge (1+\epsilon)(\tau_1+\cdots+\tau_k)$ “forces the $(k+1)$-tuple $(\tau_1,\ldots,\tau_k,\lambda_{i_{k+1}})$ to be $k$-bad, since the last entry dominates the sum of the others,” hence any $(\lambda_{i_1},\ldots,\lambda_{i_k},\lambda_{i_{k+1}})$ with $\lambda_{i_j}\le \tau_j$ is also $k$-bad.

Rather than summing over a region in $(0,\infty)^{k+1}$, the argument is organized by “fix[ing] upper bounds $\tau_j$ for $\lambda_{i_j}$, $1\le j\le k$, and then estimat[ing] the tail sum over those $i_{k+1}$ with $\lambda_{i_{k+1}} \ge (1+\epsilon)(\tau_1+\cdots+\tau_k)$.” To prove the main theorem, the proof then “uses the method of non-stationary phase,” and (per the Organization paragraph) establishes “rapid decay of the high-frequency tail in the regime where the $(k+1)$-gon inequality fails, via spectral projectors and a non-stationary phase argument.” A flat-torus example is recorded showing the frequency threshold is “essentially sharp.”

Existential–Universal
Inequality or Bound

Question: Let \(\{e_i\}\) be an \(L^2\)-orthonormal family of eigenfunctions with corresponding eigenvalues (frequencies) \(\lambda_i\), and write \(\langle f,g\rangle\) for the \(L^2\) inner product. Fix an integer \(k\ge 1\). For positive numbers \(\tau_1,\dots,\tau_k\), suppose the indices \(i_1,\dots,i_k\) satisfy \(\lambda_{i_j}\le \tau_j\) for each \(1\le j\le k\). Under these assumptions, which quantitative estimate holds for the tail of the harmonic expansion coefficients of the product \(e_{i_1}e_{i_2}\cdots e_{i_k}\)?

For every \(\epsilon>0\) and every integer \(N>0\), there exists a constant \(C_{\epsilon,N}\) such that
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\tau_2+\cdots+\tau_k)} \bigl|\langle e_{i_1}e_{i_2}\cdots e_{i_k},\, e_{i_{k+1}}\rangle\bigr|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}.
\]

Choices:
B.
For every \(\epsilon>0\) and every integer \(N>0\), there exists a constant \(C_{\epsilon,N}\) such that
\[
\sum_{\lambda_{i_{k+1}}\ge (\tau_1+\tau_2+\cdots+\tau_k)+\epsilon} \bigl|\langle e_{i_1}e_{i_2}\cdots e_{i_k},\, e_{i_{k+1}}\rangle\bigr|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}.
\]
C.
For every \(\epsilon>0\), there exists a constant \(C_{\epsilon}\) such that
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\tau_2+\cdots+\tau_k)} \bigl|\langle e_{i_1}e_{i_2}\cdots e_{i_k},\, e_{i_{k+1}}\rangle\bigr|^2
\le C_{\epsilon}.
\]
D.
There exists an integer \(N>0\) such that for every \(\epsilon>0\) there is a constant \(C_{\epsilon,N}\) with
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\tau_2+\cdots+\tau_k)} \bigl|\langle e_{i_1}e_{i_2}\cdots e_{i_k},\, e_{i_{k+1}}\rangle\bigr|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}
\]
for all positive \(\tau_1,\dots,\tau_k\) and all indices \(i_1,\dots,i_k\) with \(\lambda_{i_j}\le \tau_j\).
E.
For every \(\epsilon>0\) and every integer \(N>0\), there exists a constant \(C_{\epsilon,N}\) such that
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)\max\{\tau_1,\tau_2,\dots,\tau_k\}} \bigl|\langle e_{i_1}e_{i_2}\cdots e_{i_k},\, e_{i_{k+1}}\rangle\bigr|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}.
\]

multiplicative tail threshold replaced by additive offset

dropped rapid-decay power saving in \((\tau_1+\cdots+\tau_k)\)

quantifier order on N weakened from arbitrary N to existence of one N

sum-of-sides bad-region threshold replaced by max-side threshold


Question: Let \(\{e_i\}\) be an \(L^2\)-orthonormal basis of eigenfunctions with corresponding frequencies \(\lambda_i\), and let \(\langle \cdot,\cdot\rangle\) denote the \(L^2\) inner product. Which uniform decay statement holds for the high-frequency tail of the harmonic expansion coefficients of a product \(e_{i_1}e_{i_2}\cdots e_{i_k}\)?

For every \(\epsilon>0\) and every integer \(N>0\), there exists a constant \(C_{\epsilon,N}\) such that for every choice of positive numbers \(\tau_1,\tau_2,\ldots,\tau_k\), and for every indices \(i_1,\ldots,i_k\) satisfying \(\lambda_{i_j}\le \tau_j\) for \(1\le j\le k\), one has
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\tau_2+\cdots+\tau_k)} \left|\langle e_{i_1}e_{i_2}\cdots e_{i_k},e_{i_{k+1}}\rangle\right|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}.
\]

Choices:
B.
For every \(\epsilon>0\) and every integer \(N>0\), there exists a constant \(C_{\epsilon,N}\) such that for every choice of positive numbers \(\tau_1,\tau_2,\ldots,\tau_k\), and for every indices \(i_1,\ldots,i_k\) satisfying \(\lambda_{i_j}\le \tau_j\) for \(1\le j\le k\), one has
\[
\sum_{\lambda_{i_{k+1}}\ge \tau_1+\tau_2+\cdots+\tau_k+\epsilon} \left|\langle e_{i_1}e_{i_2}\cdots e_{i_k},e_{i_{k+1}}\rangle\right|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}.
\]
C.
For every \(\epsilon>0\), there exists a constant \(C_{\epsilon}\) such that for every choice of positive numbers \(\tau_1,\tau_2,\ldots,\tau_k\), and for every indices \(i_1,\ldots,i_k\) satisfying \(\lambda_{i_j}\le \tau_j\) for \(1\le j\le k\), one has
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\tau_2+\cdots+\tau_k)} \left|\langle e_{i_1}e_{i_2}\cdots e_{i_k},e_{i_{k+1}}\rangle\right|^2
\le C_{\epsilon}.
\]
D.
For every \(\epsilon>0\) and every integer \(N>0\), there exists a constant \(C_{\epsilon,N}\) such that for every choice of positive numbers \(\tau_1,\tau_2,\ldots,\tau_k\), and for every indices \(i_1,\ldots,i_k\) satisfying \(\lambda_{i_j}\le \tau_j\) for \(1\le j\le k\), one has
\[
\sup_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\tau_2+\cdots+\tau_k)} \left|\langle e_{i_1}e_{i_2}\cdots e_{i_k},e_{i_{k+1}}\rangle\right|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}.
\]
E.
There exists \(\epsilon>0\) such that for every integer \(N>0\) there exists a constant \(C_{\epsilon,N}\) with the following property: for every choice of positive numbers \(\tau_1,\tau_2,\ldots,\tau_k\), and for every indices \(i_1,\ldots,i_k\) satisfying \(\lambda_{i_j}\le \tau_j\) for \(1\le j\le k\), one has
\[
\sum_{\lambda_{i_{k+1}}\ge (1+\epsilon)(\tau_1+\tau_2+\cdots+\tau_k)} \left|\langle e_{i_1}e_{i_2}\cdots e_{i_k},e_{i_{k+1}}\rangle\right|^2
\le C_{\epsilon,N}(\tau_1+\tau_2+\cdots+\tau_k)^{-N}.
\]

multiplicative tail threshold above the sum

conversion from pointwise coefficient decay plus Weyl counting to an \(\ell^2\)-tail bound