[\cite{Cho}]\label{problem:Cho}
    For $\eta  \leqslant  \theta$, give a new definition of the plactic skew Schur $P$-function $\sPlacticSchur_{\theta/\eta}$
 so that it belongs to the ring generated by plactic Schur $P$-functions and it describes
the multiplicities of the ordinary $P_\lambda$ in the expansion of $P_{\theta/\eta}$ in a nice way.

\begin{problem}[\cite{Cho}]\label{problem:Cho}
    For $\eta  \leq  \theta$, give a new definition of the plactic skew Schur $P$-function $\sPlacticSchur_{\theta/\eta}$
 so that it belongs to the ring generated by plactic Schur $P$-functions and it describes
the multiplicities of the ordinary $P_\lambda$ in the expansion of $P_{\theta/\eta}$ in a nice way.
\end{problem}

The \emph{Schur functions} $s_\lambda$ are a basis of the ring of symmetric functions and their structure coefficients $c_{\lambda,\mu}^\nu$, appearing in the expansions
\[
s_\lambda \cdot s_\mu = \sum_\nu c_{\lambda,\mu}^\nu s_\nu,
\]
are the \emph{Littlewood--Richardson coefficients}. These objects play important roles in the linear representation theory of symmetric groups (where $s_\lambda$ represents a \emph{Specht module}), in the linear representation theory of general linear groups (where $s_\lambda$ represents a \emph{Schur module}), and in the Schubert calculus of Grassmannians (where $s_\lambda$ represents a \emph{Schubert cycle}). The Littlewood--Richardson coefficients may be studied combinatorially through the interlocking machinery of \emph{jeu de taquin} on \emph{Young tableaux}, the \emph{Robinson--Schensted--Knuth} (RSK) insertion algorithm, and the \emph{plactic monoid}. For exposition of these now-standard ideas, see, for example, the textbooks \cite{Fulton:YT,Manivel}.

The ideas discussed so far may all be considered ``type A.'' In other classical types, the corresponding theories are more complicated and the relations less well developed. The analogues of the usual Schur functions are the \emph{Schur $P$-functions} and \emph{Schur $Q$-functions}, which play similar roles for the projective representation theory of symmetric groups as well as for the Schubert calculus of Lagrangian and maximal orthogonal Grassmannians.

In this setting, \emph{Sagan--Worley} \cite{Sagan,Worley} insertion is an analogue of RSK that is closely connected to a \emph{Sagan--Worley} jeu de taquin. This Sagan--Worley theory does not appear to have a corresponding analogue of the plactic monoid. On the other hand, \emph{mixed} insertion \cite{Haiman} is a different analogue of RSK insertion with a corresponding \emph{shifted plactic monoid} \cite{Serrano}. This latter theory has not been known to have a corresponding jeu de taquin. We provide\footnote{The results of this paper were first announced in the extended abstract \cite{EstupinanSalamanca.Pechenik:FPSAC}.} such a jeu de taquin in \cref{sec:mixedJeudeTaquin}. Our new jeu de taquin computes mixed insertions, despite lacking some of the properties of ordinary jeu de taquin, such as confluence. There is precedent for useful theories of jeu de taquin without confluence, such as the $K$-theoretic jeu de taquin theories of \cite{Thomas.Yong:K, Pechenik.Yong} and we hope that our mixed jeu de taquin will lead to similar such developments.

In type A, the combinatorics of Schur functions can realized through \emph{plactic skew Schur functions}, which plactically encode both jeu de taquin rectification and RSK insertion. This realization yields an algebraic perspective on these combinatorial theories, as well as on the Littlewood--Richardson coefficients. Serrano \cite[p.~379]{Serrano} introduced a notion of \emph{plactic skew Schur $P$-functions}, which he conjectured \cite[Conjecture~2.12]{Serrano} (see also, \cite[Conjecture~7.8]{Cho}) would play an analogous role in other classical types. If true, this would have given an algebraic substitute for the missing jeu de taquin associated to mixed insertion and to the shifted plactic monoid. Unfortunately, Serrano's conjecture was not correct. Precisely, Cho \cite[\textsection 7]{Cho} gave explicit examples to show that Serrano's plactic skew Schur $P$-functions do not lie in the desired ring and hence they cannot serve the desired role in substituting for jeu de taquin or in shedding algebraic light on structure coefficients. Having disproved Serrano's conjecture, Cho asks in \cite[Open Problem~7.12(1)]{Cho} for a new definition that would accomplish Serrano's goals. More precisely, she sets the following task:

The \emph{Schur functions} $s_\lambda$ are a basis of the ring of symmetric functions and their structure coefficients $c_{\lambda,\mu}^\nu$, appearing in the expansions
\[
s_\lambda \cdot s_\mu = \sum_\nu c_{\lambda,\mu}^\nu s_\nu,
\]
are the \emph{Littlewood--Richardson coefficients}. These objects play important roles in the linear representation theory of symmetric groups (where $s_\lambda$ represents a \emph{Specht module}), in the linear representation theory of general linear groups (where $s_\lambda$ represents a \emph{Schur module}), and in the Schubert calculus of Grassmannians (where $s_\lambda$ represents a \emph{Schubert cycle}). The Littlewood--Richardson coefficients may be studied combinatorially through the interlocking machinery of \emph{jeu de taquin} on \emph{Young tableaux}, the \emph{Robinson--Schensted--Knuth} (RSK) insertion algorithm, and the \emph{plactic monoid}. For exposition of these now-standard ideas, see, for example, the textbooks \cite{Fulton:YT,Manivel}.

In this setting, \emph{Sagan--Worley} \cite{Sagan,Worley} insertion is an analogue of RSK that is closely connected to a \emph{Sagan--Worley} jeu de taquin. This Sagan--Worley theory does not appear to have a corresponding analogue of the plactic monoid. On the other hand, \emph{mixed} insertion \cite{Haiman} is a different analogue of RSK insertion with a corresponding \emph{shifted plactic monoid} \cite{Serrano}. This latter theory has not been known to have a corresponding jeu de taquin. We provide\footnote{The results of this paper were first announced in the extended abstract \cite{EstupinanSalamanca.Pechenik:FPSAC}.} such a jeu de taquin in \cref{sec:mixedJeudeTaquin}. Our new jeu de taquin computes mixed insertions, despite lacking some of the properties of ordinary jeu de taquin, such as confluence. There is precedent for useful theories of jeu de taquin without confluence, such as the $K$-theoretic jeu de taquin theories of \cite{Thomas.Yong:K, Pechenik.Yong} and we hope that our mixed jeu de taquin will lead to similar such developments.

\medskip
{\bf This paper is structured as follows.} In \cref{sec:background}, we recall background and give necessary new definitions. Our treatment differs notationally from standard approaches in ways that we believe are clarifying. In \cref{sec:mixedJeudeTaquin}, we introduce mixed jeu de taquin and prove various properties of it. In particular, we show in \cref{thm:mixedRectification} that it computes Haiman's mixed insertions. \cref{sec:Cho} gives a new definition of plactic skew Schur $P$-functions $\sPlacticSchur_{\theta/\eta}$ and shows that these give a solution to \cref{problem:Cho}.

A \newword{hook subword} of a word $w$ in alphabet $\cN$ is a subword $d\cdot i$ such that $d$ is strictly decreasing and $i$ is weakly increasing. If $w$ is a hook subword of itself, we say $w$ is a \newword{hook word}. For example, $4211688$ is a hook word.
 Let $\lambda$ be a strict partition. Denote by $\hookword(\lambda)$ the set of words $w=w_1\ldots w_k$, where each $w_i$ is a hook word of length $\lambda_{\ell(\lambda)-i+1}$ and such that, for all $i>1$, $w_i$ is a longest hook subword of $w_{i-1}w_i$.
 Then the \newword{shifted free Schur function} of shape $\lambda$ is the formal power series in noncommuting variables
\[\sFreeSchur_\lambda \coloneqq \sum_{ w\: : \: w\in \hookword(\lambda) } w \in \Z\npowerser{x_1, x_2, \dots}.\]
(Shifted free Schur functions were first introduced by Serrano \cite{Serrano} using a definition based on \emph{mixed reading words} for certain tableaux; our definition, first stated in \cite{EstupinanSalamanca.Pechenik}, is different from his, but is equivalent.) Note that shifted free Schur functions are not being defined in the skew setting; this fact is the crux of Cho's open problem.

The \newword{shifted plactic monoid} \cite{Serrano} is the quotient $\sPlactic$ of the free monoid on $\cN$ by the eight families of relations
 \begin{align*}
 &abdc\sim adbc \quad \text{for all } a\leq b \leq c<d; \\
 &acdb\sim acbd \quad \text{for all } a\leq b < c \leq d; \\
 &dacb\sim adcb \quad \text{for all } a\leq b < c <d; \\
 &badc\sim bdac \quad \text{for all } a< b \leq c<d; \\
 &cbda\sim cdba \quad \text{for all } a< b <c \leq d; \\
 &dbca\sim bdca \quad \text{for all } a< b \leq c<d; \\
 &bcda\sim bcad \quad \text{for all } a< b \leq c\leq d; \\
 &cadb\sim cdab \quad \text{for all } a\leq b < c\leq d.
 \end{align*}
The \newword{shifted plactic Schur function} $\sPlacticSchur_\lambda$ is the image of the shifted free Schur function $\sFreeSchur_\lambda$ under the projection map identifying monomials whose sequences of variable subscripts are equivalent in the shifted plactic monoid. Then $\sPlacticSchur_\lambda \in \Z \llbracket \sPlactic \rrbracket$, the quotient of $\Z\npowerser{x_1, x_2, \dots}$ by the shifted plactic relations on variables. By further projecting to the ring in commuting variables, we recover the ordinary Schur $P$-functions.

\begin{definition}\label{def:skewplacticS}
The \newword{skew plactic Schur $P$-function} is
 \[ \sPlacticSchur_{\nu/\mu}\coloneqq \frac{1}{2^{\diag(\nu/\mu)}} \sum_{T\in \shSSYTprime(\nu/\mu)} [F\circ \rectify(T)]_\sPlactic \in \Z \llbracket \sPlactic \rrbracket, \]
where $F$ is the map that raises all diagonal positions, $\rectify(\cdot)$ signifies modified Sagan--Worley rectification, and $\diag(\nu/\mu)$ denotes the number of diagonal positions in the skew shape $\nu/\mu.$ (Note that $\diag(\nu / \mu) = \ell(\nu) - \ell(\mu)$.)
\end{definition}

We need some preliminary results clarifying Sagan--Worley's rectification on skew semistandard tableaux which may contain low entries on the diagonal. While it is well known that rectifying shifted standard skew tableaux to a shifted standard target yields $b_{\lambda,\mu}^\nu$ tableaux regardless of the target chosen, the analogous result is not as well explained in the literature for the more general case. Essentially, \cref{thm:generalizedSaganWorley} and \cref{lem:WorleyRectCondition} can be found in \cite{Sagan} and \cite{Worley}, respectively; but the proof in \cite{Worley} is difficult to retrieve and it seems that \cref{lem:WorleyRectCondition} is not used for our same purposes in \cite{Sagan}. These results are also stated in \cite[Proposition~7.3]{Cho} without proof. For the sake of completeness, we include proofs of those results below and use them to show that our definition of shifted plactic skew Schur $P$-functions satisfies Cho's requirements.

The following shows that our definition solves Cho's Open Problem:
\begin{thm}
 Let $\mu<\nu$ be strict partitions. Then $\sPlacticSchur_{\nu/\mu} \in \mathbb{Q}[\sPlacticSchur_\lambda]_\lambda$ and
 \[\sPlacticSchur_{\nu/\mu} = \sum_{\lambda} \frac{2^{\ell(\lambda)}}{2^{\diag(\nu/\mu)}} b_{\lambda,\mu}^\nu \sPlacticSchur_\lambda. \]
 In particular, the expansion coefficients of $\sPlacticSchur_{\nu/\mu}$ in the plactic Schur $P$-basis are equal to the expansion coefficients of the ordinary skew Schur $P$-function $P_{\nu/\mu}$ in the ordinary Schur $P$-basis.
\end{thm}
\begin{proof}
 By \cref{def:skewplacticS}, we need to prove that for any semistandard tableau $S \in \shSSYTprime(\lambda)$, the number of skew tableaux $T \in \shSSYTprime(\nu/\mu)$ with $[F\circ \rectify(T)]_\sPlactic = [F(S)]_\sPlactic$ is equal to $2^{\ell(\lambda)}b_{\lambda,\mu}^\nu$. Noting that $b_{\lambda,\mu}^\nu$ is the number of skew tableaux $T \in \shSSYTprime(\nu/\mu)$ rectifying to any semistandard tableau $U \in \shSSYTprime(\lambda)$ by \cref{thm:generalizedSaganWorley}, it suffices to establish that there are $2^{\ell(\lambda)}$ semistandard tableaux $\hat{S} \in \shSSYTprime(\lambda)$ with $F(\hat{S})=F(S).$

We solve this problem. Interestingly, the key tool of our solution is the Sagan--Worley jeu de taquin theory. By combining this jeu de taquin with shifted plactic theory, we obtain plactic skew Schur $P$-functions that lie in the correct ring and that govern the expansion multiplicities of ordinary Schur $P$-functions, as desired.

\medskip
{\bf This paper is structured as follows.} In \cref{sec:background}, we recall background and give necessary new definitions. Our treatment differs notationally from standard approaches in ways that we believe are clarifying. In \cref{sec:mixedJeudeTaquin}, we introduce mixed jeu de taquin and prove various properties of it. In particular, we show in \cref{thm:mixedRectification} that it computes Haiman's mixed insertions. \cref{sec:Cho} gives a new definition of plactic skew Schur $P$-functions $\sPlacticSchur_{\theta/\eta}$ and shows that these give a solution to \cref{problem:Cho}.

To solve \cref{problem:Cho}, the paper uses “the Sagan--Worley jeu de taquin theory” as the “key tool.” “By combining this jeu de taquin with shifted plactic theory,” it obtains plactic skew Schur $P$-functions that “lie in the correct ring” and “govern the expansion multiplicities of ordinary Schur $P$-functions, as desired.” The structure indicated is: background/definitions (\cref{sec:background}); introduction of mixed jeu de taquin and its properties (\cref{sec:mixedJeudeTaquin}), including showing it computes Haiman’s mixed insertions via \cref{thm:mixedRectification}; then \cref{sec:Cho} “gives a new definition” of $\sPlacticSchur_{\theta/\eta}$ and shows these solve \cref{problem:Cho}.

To solve the main problem, the paper uses “the Sagan--Worley jeu de taquin theory” as the “key tool.” “By combining this jeu de taquin with shifted plactic theory,” it obtains plactic skew Schur $P$-functions that “lie in the correct ring” and “govern the expansion multiplicities of ordinary Schur $P$-functions, as desired.” The structure indicated is: background/definitions (\cref{sec:background}); introduction of mixed jeu de taquin and its properties (\cref{sec:mixedJeudeTaquin}), including showing it computes Haiman’s mixed insertions via the following theorem.

\begin{theorem}\label{thm:mixedRectification}
        Let $T$ be an arbitrary shifted Young tableau and $\unpr y$ a high letter. Then performing the mixed insertion of $\unpr y$ on $T$ results in the tableau
        $\mathrm{rect}_\mix \left( T\oplus \unpr y \right)$.
    \end{theorem}

Then \cref{sec:Cho} “gives a new definition” of $\sPlacticSchur_{\theta/\eta}$ and shows these establish the main problem.

[Cho]
    For $\eta  \leqslant  \theta$, give a new definition of the plactic skew Schur $P$-function $\sPlacticSchur_{\theta/\eta}$
 so that it belongs to the ring generated by plactic Schur $P$-functions and it describes
the multiplicities of the ordinary $P_\lambda$ in the expansion of $P_{\theta/\eta}$ in a nice way.

Question: Let \(\eta \leq \theta\) be strict partitions, so that \(\theta/\eta\) is a shifted skew shape. Here \(P_{\theta/\eta}\) denotes the ordinary skew Schur \(P\)-function, and the plactic Schur \(P\)-functions are the generators of the corresponding plactic ring. Which statement gives the required property of a new definition of the plactic skew Schur \(P\)-function \(\sPlacticSchur_{\theta/\eta}\)?

One seeks a definition of \(\sPlacticSchur_{\theta/\eta}\) such that \(\sPlacticSchur_{\theta/\eta}\) lies in the ring generated by the plactic Schur \(P\)-functions and, moreover, it describes in a natural way the multiplicities of the ordinary \(P_\lambda\) appearing in the expansion of \(P_{\theta/\eta}\).

Choices:
B. One seeks a definition of \(\sPlacticSchur_{\theta/\eta}\) such that \(\sPlacticSchur_{\theta/\eta}\) itself is a single plactic Schur \(P\)-function, and, moreover, its coefficients directly equal the Littlewood--Richardson-type structure constants for the product of plactic Schur \(P\)-functions.
C. One seeks a definition of \(\sPlacticSchur_{\theta/\eta}\) such that \(\sPlacticSchur_{\theta/\eta}\) lies in the ring generated by the plactic Schur \(P\)-functions.
D. One seeks a definition of \(\sPlacticSchur_{\theta/\eta}\) such that \(\sPlacticSchur_{\theta/\eta}\) lies in the ring generated by the plactic Schur \(P\)-functions and, moreover, it computes in a natural way the multiplicities of the ordinary \(P_\lambda\) appearing in the product expansion of \(P_\theta P_\eta\).
E. One seeks a definition of \(\sPlacticSchur_{\theta/\eta}\) such that, for every pair of strict partitions \(\eta \leq \theta\), there is a canonical expression of \(\sPlacticSchur_{\theta/\eta}\) as a polynomial with nonnegative integer coefficients in the plactic Schur \(P\)-functions, and these coefficients coincide with the multiplicities of the ordinary \(P_\lambda\) in \(P_{\theta/\eta}\).

membership-in-generated-ring vs single-basis-element / expansion-multiplicity role

dropped the requirement about describing multiplicities of ordinary \(P_\lambda\) in \(P_{\theta/\eta}\)

skew-expansion multiplicities replaced by product-expansion multiplicities

unjustified strengthening to canonical polynomial with nonnegative integer coefficients

The stem gives background terminology but does not state the target property itself. The correct answer is not explicitly revealed, and the alternatives are close enough that the answer is not leaked trivially.

The item mainly asks the test-taker to identify the precise stated property of the new definition, which is essentially theorem/definition recall rather than a nontrivial mathematical conclusion derived from premises.

Some reasoning is needed to separate the exact claim from weaker, stronger, or confused variants (especially C, D, and E), but the task is primarily recognition of the intended statement rather than genuine generative mathematical reasoning.

The distractors are plausible and well-targeted: one is too weak, one confuses skew-expansion with product-expansion, one overstrengthens the claim, and one changes the object entirely. These reflect realistic failure modes.

A reasonably well-constructed recall/discrimination MCQ with strong distractors and little answer leakage, but it is largely a statement-identification item rather than a genuinely generative reasoning question.


In statistical hypothesis testing the objective is to decide if a null hypothesis $H_0$, which specifies a distribution or statistical model for random data $X$, should be retained or rejected in favor of alternative hypothesis $H_1$ based on realization $X = x$. The decision can be denoted by $\delta(x)\in\{0,1\}$, where $\delta(x) =$ 0 (1) means that $H_0$ is retained (rejected). The Neyman-Pearson (NP) paradigm stipulates that $\delta$ should maximize power subject to the constraint that the type 1 error rate is not more than some prespecified level $\alpha$.
Formally, $\delta$ should maximize $E_1[\delta(X)]$ subject to the constraint that $E_0[\delta(X)]\leq \alpha$, where $E_j$ denotes an expectation taken under $H_j$. For example, in the simple versus simple setting $H_0:X\sim P_0$ is tested against $H_1:X\sim P_1$, and the most powerful level $\alpha$ decision rule is $\delta(x) = I(\lambda(x)\geq c)$ where $I(\cdot)$ is the indicator function, $\lambda(x) = \frac{p_1(x)}{p_0(x)}$ is the likelihood ratio statistic, $p_j$ is the probability mass function for $P_j$ under $H_j$, and $c$ is chosen so that $E_0[\delta(X)] = \alpha$ \citep{Neyman1933}. The corresponding $p$-value can be computed via $p(x) = \Pr_0\{\lambda(X)\geq \lambda(x)\}$.

However, if $X$ has a discrete distribution, it is not generally possible to define $\delta(X)$ such that $E_0[\delta(X)]=\alpha$, and several different types of $p$-values can be considered. Let us consider a simple example to facilitate discussion.

Let us pause to consider a concrete example. Table \ref{ExampleTable} provides an illustration of MD natural $p$-values that agree with the natural $p$-values from the usual LRT in Example \ref{BernoulliTestFunction}.
\begin{table}[ht]
\caption{\label{ExampleTable} The first 5 columns list an $x \in \mathcal{X}$ in Example \ref{BernoulliExample}. The following columns contain $p_0(x)$, $p_1(x)$, likelihood ratio $\lambda(x)$, a ranking $R(x)$ that agrees with $\lambda(x)$, and natural MD $p$-values based on $R(x)$ computed $P^{MD-\lambda}(x,1) = \Pr_0\{R(X)\leq R(x)\}$ and natural $p$-value based on the likelihood ratio statistic $P^\lambda(x,1) = \Pr_0\{\lambda(X)\geq \lambda(x)\}$}
\begin{tabular}{cccccccccll}
$x_1$&$x_2$&$x_3$&$x_4$&$x_5$&$p_0(x)$ & $p_1(x)$ & $\lambda(x)$ & $R(x)$ & $P^{MD}(x,1)$& $P^\lambda(x,1)$\\ \hline
1 & 1 & 1& 1& 1 & 0.03125 & 0.32768 & 10.4857 & 1 & 0.03125 & 0.03125\\
0 & 1 & 1& 1& 1 & 0.03125 & 0.08192 & 2.62144 & 2 & 0.06250 & 0.1875 \\
1 & 0 & 1& 1& 1 & 0.03125 & 0.08192 & 2.62144 & 3 & 0.09375& 0.1875\\
1 & 1 & 0& 1& 1 & 0.03125 & 0.08192 & 2.62144 & 4 & 0.125& 0.1875\\
1 & 1 & 1& 0& 1 & 0.03125 & 0.08192 & 2.62144 & 5 & 0.15625& 0.1875 \\
1 & 1 & 1& 1& 0 & 0.03125 & 0.08192 & 2.62144 & 6 & 0.1875& 0.1875\\
0 & 0 & 1& 1& 1 & 0.03125 & 0.02048 & 0.65536 & 7 & 0.21875&0.5\\
1 & 0 & 0& 1& 1 & 0.03125 & 0.02048 & 0.65536 & 8 & 0.25&0.5\\
\end{tabular}
\end{table}
Observe that $R$ agrees with $\lambda$ since $\lambda(x)\geq\lambda(y)$ implies $R(x)\leq R(y)$ for each $x$,$y$ listed. We further observe that $R$ is one-to-one since each $x$ is assigned a unique ranking for the support points depicted. Hence, the resulting $p$-value $P^{MD-\lambda}(x,1)$ is minimally discrete. Less technically, the terminology ``minimal'' emphasizes that among all possible other natural $p$-values based on the likelihood ratio test, this $p$-value has as many support points as possible (32 in this example), and hence minimizes discreteness effect. The p-value based on the LRT is not minimal since it only has 6 support points when 32 are possible.

Next let us compare MR-$T$ $p$-values to randomized $p$-values based on $T$ that need not be MR. Claim (C5) stipulates that both $p$-values are uniformly distributed under the null hypothesis. Claim (C6) states that if $T$ is a sufficient statistic, then both $p$-value statistics are equal in distribution whether the null hypothesis is true or false. This is perhaps not surprising given that most powerful size $\alpha$ decision rules are not unique in discrete testing (recall the Introduction) and both tests are size $\alpha$ tests based on sufficient statistics. However, the MR $p$-value dominates its non-MR counterpart in that the generation of $u$ has ``minimal impact'' on $p(x,u)$. In particular, Claim (C7) states that the variance attributable to the auxiliary $U$ is smaller for the MR-$T$ $p$-value if $T$ is sufficient. Consequently, variation from replication to replication that is attributable to generation of $U$ is minimized. In this sense, we may say that the replicability of $p^{MR-T}(X,U)$ is greater.
\begin{theorem}[MR-$T$ versus $T$: Randomized p-values and decision functions] \label{Thm2}
Let decision functions and p-values be defined as Definitions \ref{T} and \ref{MDT} with $U$ being uniformly distributed over the unit interval and independent of $X$. Then claim (C5) below is true. If additionally $T$ is a sufficient statistic, then claim (C6) and (C7) are true:
\begin{enumerate}
\item[(C5)] $E_0[\delta_\alpha^{MR-T}(X,U)] = E_0[\delta_\alpha^{T}(X,U)] = \alpha$ $\forall \alpha\in[0,1]$ and hence \\
$Pr_0\{\Pr(P^{MR-T}(X,U)\leq t\} = \Pr_0\{P^{T}(X,U)\leq t\} = t$ $\forall t \in [0,1]$,\\
\item[(C6)] $E_\theta[\delta_\alpha^{MR-T}(X,U)] = E_\theta[\delta_\alpha^T(X,U)]$ $\forall \alpha\in[0,1],\theta\in\Theta$ \\
and hence $Pr_\theta\{\Pr(P^{MR-T}(X,U)\leq t\} = \Pr_\theta\{P^{T}(X,U)\leq t\}$ $\forall t\in[0,1], \theta\in\Theta$, \\
\item[(C7)] $Var_\theta(P^{MD-T}(X,U)|X)\leq Var_\theta(P^{T}(X,U)|X)$ $\forall \theta\in\Theta$.
\end{enumerate}
\end{theorem}

A brief discussion regarding the potential impact of results here is warranted. First consider multiple hypothesis testing. A multiple hypothesis testing (MTP) procedure tests null hypotheses $H_{01},H_{02},..., H_{0M}$ with $p$-values $P_1, P_2, ..., P_M$, which are often assumed to be uniformly distributed under the null hypotheses. Formally, they are defined $\delta_i = I(P_i\leq t)$ for $i = 1, 2, ..., M$ and some $t\in[0,1]$. For example, a Bonferroni procedure chooses $t = \alpha/M$ while the \cite{Benjamini1995} procedure for False Discovery Rate control chooses $\hat{t} = \sup\{s:\frac{s}{\max\{\sum_i I(P_i\leq s),1\}}\leq \alpha\}$ (with the supremum of the empty set defined to be 0), which can be shown to converge to some constant $t$ as $M\rightarrow \infty$ with probability 1 under weak dependence (cf. \cite{Genovese2002,Storey2004, Habiger2015}). While procedures are often studied under the assumption that $\Pr_0(P_i\leq t) = t$, this assumption can be relaxed to $\Pr_0(P_i\leq t)\geq t$. Claim (C4) in Theorem \ref{Thm1} ensures that, if opting for natural $p$-values, the MD-natural $p$-values are still valid since they satisfy the later relaxed assumption. Claims (C1) and (C3) give that they result in more power. If opting for randomized $p$-values then both MR $p$-values and their non-minimal counterparts are uniformly distributed under the null hypotheses. If randomized $p$-values are functions of sufficient statistics, then we should not anticipate any more or less power if opting for their MR counterparts (claim (C6)), as the Neyman Pearson Lemma suggests. However, claim (C7) stipulates that fewer discoveries will rely on the generation of $u$ if utilizing the MR $p$-values. To see this, consider Example \ref{BernoulliTestFunction} and consider implementing a level $\alpha = 0.05$ test. In this example, the decision to reject or retain $H_0$ will depend upon $U$ with probability $\Pr\{T(X) = 4\}$, which is 5 times more likely than its MR counterpart that only utilizes $u$ for one $x\in [T(x)=4]$.

\section{False Discovery Rate Methods and MD $p$-values}
Next consider testing null hypotheses $H_m:\theta_m = 0$ with data $X_m\in\mathcal{X}_m$, where $\mathcal{X}_m$ is the countable support of $X_m$, for $m = 1, 2, ..., M$. A size $\alpha$ test function and its corresponding decision function and $p$-value are defined as in Section 2 but indexed by $m$. Specifically $\phi_{\alpha,m}(x_m)$ is nondecreasing and right continuous in $\alpha$ and satisfies $E_{0}[\phi_{\alpha,m}(X_m)] = \alpha$, where $E_0$ denotes an expectation taken over $X_m$ under $H:\theta_m = 0$. Likewise, $\delta_{\alpha,m}(x_m,u_m) = I(u_m\leq \phi_{\alpha,m}(x_m))$ and $P_m(x_m, u_m) = \inf\{\alpha:\delta_{\alpha,m}(x_m,u_m) = 1\}$. We shall write $P_m = P_m(x_m,u_m)$ for short whenever possible in this section and let $P$ denote a collection of $p$-values.

\noindent Observe that $\lambda(x) = p_{1}(x)/p_{0}(x)$ is increasing in $T(x)$ so the likelihood ratio test rejects $H_0$ for large values of $T(x)$. However, observe $T(x)$ takes values in $\{0,1,...,5\}$ and that $\Pr_0\{T(X) \geq 5\} = 0.03125$ but $\Pr_0\{T(X)\geq 4\} = 0.1875$.  Hence, it is not possible to define a $\delta$, i.e. choose $c$, such that $E_0[\delta(X)] = \Pr_0\{T(X)\geq c\} = \alpha$ if, for example, $\alpha = 0.1$.  Randomized testing \citep{Pearson1950, Tocher1950} would be necessary. In Example \ref{BernoulliExample}, a most powerful level $0.1$ test function is
\begin{equation}\label{BernoulliTestFunction}
\phi(x) = \left\{\begin{array}{rcl} 1 & \mbox{if} &T(x) > 4 \\
 0.44 & \mbox{if} & T(x) = 4  \\
 0 & \mbox{if}& T(x) < 4.
 \end{array}\right.
\end{equation}
where $\phi(x)$ represents the rejection probability. See, for example, \cite{Lehmann1993}. The randomized test, when implemented, can be written as a randomized decision rule $\delta(x,u) = I(u\leq \phi(x))$ where $u$ is a realization from a standard uniform distribution. The corresponding randomized $p$-value can be computed $p(x,u) =  \Pr_{0}\{T(X)>T(x)\}+ u\Pr_{0}\{T(X) = T(x)\}$.  See, for example, \cite{Pena2011, Habiger2011}.   In practice, we may also report the corresponding natural $p$-value: $p(x,1)=\Pr_{0}\{T(X)\geq T(x)\}$ (cf. \cite{Casella2002}), mid-p-value \citep{Lan61}: $p(x,1/2) = \Pr_{0}\{T(X)>T(x)\}+ 1/2\Pr_{0}\{T(X) = T(x)\}$, or even the fuzzy $p$-value \citep{GeyMee05}, which in this example amounts to reporting the interval $[p(x,0), p(x,1)]$.  The curious reader is referred to \cite{Agr07, Wells2010} for comparison of approaches in traditional single null hypothesis testing settings. More recent research studies the impact of the employing mid-p-values, natural p-values, or randomized in downstream methods.

First let us consider randomized $p$-value literature. \cite{Habiger2011, Dickhaus2013} provided conditions that ensure randomized $p$-values are uniformly distributed under the null hypotheses so that the \cite{Benjamini1995} multiple hypothesis testing procedure, and other adaptive procedures (eg. \cite{Storey2004}) for false discovery rate (FDR) control are valid.  \cite{Ochieng2024} combined randomization methods in \cite{Hoang2022,Hoang2022b}  with aforementioned randomization methods to develop improved 2 step procedures for multiple testing of composite null hypotheses with discrete data. \cite{Habiger2015, Dai2019} demonstrated that randomized p-values can be used to develop more powerful multiple testing procedures even if (adjusted) randomized $p$-values are not ultimately reported, say as in the fuzzy BH procedure in \cite{Kulinskaya2009}. Of course, randomized $p$-values may be more variable than their non-randomized counterparts, which can be manifested as decreasing replication probability. That is, the probability of realizing the same randomized p-value in a replicated experiment is 0 if $u$ is regenerated.

Other research is aimed at establishing conservative (or less conservative) behavior for mid and natural p-values.  For example, \cite{Doehler2018, Chen2020, Chen2020b} provide upper bounds for the FDR when the BH procedure is applied to mid-p-values.   In meta analysis, \cite{Rubin-Delanchy2018} provide bounds for the distribution of Fisher's test statistic $T= -2 \sum_i \log(P_i)$ under the null hypothesis when $P_i$ represents a mid-p-value. The idea builds upon \cite{Hwang2001}, where it was shown that the mid-p-value is stochastically less than or equal to a uniform distribution in the convex order \citep{Shaked2007} under the null hypothesis.  \cite{Wang2024} also used stochastic ordering to calibrate mid-p-values into valid p-values, valid E-values \citep{Vovk2021} and Bayes Factors. Of course, established inequalities need not be sharp.

Ideally inequalities for downstream methods that utilize mid-p-values or natural p-values would be as sharp as possible and, if opting for randomized $p$-values, generated uniform variates would add minimal variation.  This motivates our minimality principle, which ensures that the aforementioned detrimental impacts when dealing with the ``discreteness'' of the data are minimized.  Specifically, we develop \textit{minimally discrete} (MD) mid-p-values,  MD natural $p$-values, and \textit{minimally randomized} (MR) p-values that dominate their traditional mid-$p$-value, natural $p$-value and randomized $p$-value counterparts, respectively. Specifically, it is shown that inequalities based on the usual stochastic order for natural $p$-values and inequalities based on the convex order for mid-$p$-values are sharpened.  Likewise, variation attributable to independeintly generated auxiliary variates for randomized $p$-values is smaller.

Our implication that ``less discrete'' and ``less randomized'' p-values are available is bold and therefore warrants a brief discussion. Reconsider Example \ref{BernoulliExample} and let us compare \begin{equation}\label{AnotherMPTest}
\phi^*(x) = \left\{\begin{array}{rcl} 1 & \mbox{if} & x\in \{ (1,1,1,1,1), (1,1,1,1,0),  (0,1,1,1,1)\}  \\
 0.2 & \mbox{if} & x=(1,1,0,1,1)  \\
 0 & & \textrm{otherwise}
 \end{array}\right.
\end{equation}
to the most powerful test $\phi(x)$ in \eqref{BernoulliTestFunction}. A curious reader may verify that the type 1 error rates are $E_0[\phi(X)] = E_0[\phi^*(X)] = 0.1$ and that the powers are $E_1[\phi(X)] = E_1[\phi^*(X)] = 0.5079$. Clearly $\phi^*$ and $\phi$ are two distinct most powerful level $0.1$ tests. In particular, both tests automatically reject $H_0$ when $T(x)>4$ and automatically retain $H_0$ when $T(x)<4$.  The difference is that $\phi(x)$ rejects $H_0$ with probability 0.44 for all five $x\in [T(x)=4]$ while $\phi^*(x)$ only randomly rejects $H_0$ for one $x\in [T(x)=4]$.  It automatically rejects $H_0$ if $x = (1,1,1,1,0)$ or $x= (0,1,1,1,1)$ and automatically retains $H_0$ when $x=(1,0,1,1,1)$ or $x = (1,1,1,0,1)$.  The key realization is that a most powerful test function can be defined ``arbitrarily on the set $[\lambda(x) = k]$'' \citep{Lehmann1997} as long as the resulting test function has expectation equal to $\alpha$ under $H_0$.  While mathematical statistics texts (cf. \cite{Cox1974, Lehmann1997}) acknowledge this point, further development is cited as beyond scope: in testing a single null hypothesis $\phi$ is sometimes viewed as a mathematical mechanism for illustrating the NP Lemma rather than a practical method.  However, as noted previously, p-values derived from test functions like $\phi$ are increasingly utilized as inputs in a downstream method.  In this setting, further research is warranted.

In Example \ref{BernoulliExample}, the argument is that the likelihood ratio \(\lambda(x)=p_1(x)/p_0(x)\) is increasing in \(T(x)\), so the likelihood ratio test rejects \(H_0\) for large values of \(T(x)\). Since \(T(x)\in\{0,1,\dots,5\}\) is discrete, one cannot generally choose a cutoff \(c\) such that \(E_0[\delta(X)]=\Pr_0\{T(X)\ge c\}=\alpha\) (e.g. \(\Pr_0\{T(X)\ge5\}=0.03125\) but \(\Pr_0\{T(X)\ge4\}=0.1875\), so \(\alpha=0.1\) cannot be matched by a nonrandomized cutoff). Therefore randomized testing is necessary, and a most powerful level \(0.1\) test is obtained by rejecting with probability 1 when \(T(x)>4\), with probability \(0.44\) when \(T(x)=4\), and with probability 0 when \(T(x)<4\), i.e. \eqref{BernoulliTestFunction}. The text further notes that the randomized test can be implemented as \(\delta(x,u)=I(u\le \phi(x))\) with \(u\sim \mathrm{Unif}(0,1)\), yielding a randomized \(p\)-value \(p(x,u)=\Pr_0\{T(X)>T(x)\}+u\Pr_0\{T(X)=T(x)\}\). It is also observed that most powerful tests need not be unique: a different most powerful level \(0.1\) test \(\phi^*\) in \eqref{AnotherMPTest} has the same type I error and power as \(\phi\), illustrating that one may define a most powerful test function "arbitrarily on the set \([\lambda(x)=k]\)" as long as its expectation under \(H_0\) equals \(\alpha\).

In the following example.

\begin{example}\label{BernoulliExample}
Let $X = (X_1, X_2, ..., X_5)$ be iid Bernoulli($\theta$) random variables with joint probability mass function $p_\theta(x) = \theta^{T(\mathbf{x})}(1-\theta)^{5-T(\mathbf{x})}$ where $T(x) = \sum_ix_i$, and consider testing null hypothesis $H_0:\theta = 0.5$ against alternative hypothesis $H_1:\theta = 0.8$.
\end{example}

the argument is that the likelihood ratio \(\lambda(x)=p_1(x)/p_0(x)\) is increasing in \(T(x)\), so the likelihood ratio test rejects \(H_0\) for large values of \(T(x)\). Since \(T(x)\in\{0,1,\dots,5\}\) is discrete, one cannot generally choose a cutoff \(c\) such that \(E_0[\delta(X)]=\Pr_0\{T(X)\ge c\}=\alpha\) (e.g. \(\Pr_0\{T(X)\ge5\}=0.03125\) but \(\Pr_0\{T(X)\ge4\}=0.1875\), so \(\alpha=0.1\) cannot be matched by a nonrandomized cutoff). Therefore randomized testing is necessary, and a most powerful level \(0.1\) test is obtained by rejecting with probability 1 when \(T(x)>4\), with probability \(0.44\) when \(T(x)=4\), and with probability 0 when \(T(x)<4\), i.e.
\begin{equation}\label{BernoulliTestFunction}
\phi(x) = \left\{\begin{array}{rcl} 1 & \mbox{if} &T(x) > 4 \\
 0.44 & \mbox{if} & T(x) = 4  \\
 0 & \mbox{if}& T(x) < 4.
 \end{array}\right.
\end{equation}
The text further notes that the randomized test can be implemented as \(\delta(x,u)=I(u\le \phi(x))\) with \(u\sim \mathrm{Unif}(0,1)\), yielding a randomized \(p\)-value \(p(x,u)=\Pr_0\{T(X)>T(x)\}+u\Pr_0\{T(X)=T(x)\}\). It is also observed that most powerful tests need not be unique: a different most powerful level \(0.1\) test \(\phi^*\) in \eqref{AnotherMPTest} has the same type I error and power as \(\phi\), illustrating that one may define a most powerful test function "arbitrarily on the set \([\lambda(x)=k]\)" as long as its expectation under \(H_0\) equals \(\alpha\).

Question: Consider a hypothesis-testing setup in which \(X=(X_1,X_2,\ldots,X_5)\) consists of five independent and identically distributed Bernoulli\((\theta)\) random variables. For \(x=(x_1,\ldots,x_5)\), define \(T(x)=\sum_{i=1}^5 x_i\), so the joint probability mass function is \(p_\theta(x)=\theta^{T(x)}(1-\theta)^{5-T(x)}\). Which hypothesis test is being considered in this model?

The null hypothesis is \(H_0:\theta=0.5\) and the alternative hypothesis is \(H_1:\theta=0.8\).

Choices:
B. The null hypothesis is \(H_0:\theta=0.8\) and the alternative hypothesis is \(H_1:\theta=0.5\).
C. The test compares two simple hypotheses for the Bernoulli parameter \(\theta\), with \(H_0:\theta=0.5\) against a fixed alternative \(H_1:\theta\neq 0.5\).
D. The null hypothesis is \(H_0:\theta\le 0.5\) and the alternative hypothesis is \(H_1:\theta>0.5\).
E. The null hypothesis is \(H_0:\theta=0.5\) and the alternative hypothesis is \(H_1:\theta\ge 0.8\).

The stem states only the Bernoulli sampling model and sufficient statistic; it does not explicitly or implicitly reveal the specific null and alternative hypotheses keyed in choice A.

This is not a direct restatement of a theorem or definition. The item asks for identification of a testing setup rather than merely rephrasing a stated conclusion.

There is essentially no valid generative reasoning path from the stem to the keyed answer, because the hypotheses are not determined by the model description alone. The item tests guessing or hidden-context recall rather than mathematical reasoning.

The distractors are mathematically plausible and reflect common confusions (reversing null/alternative, simple vs. composite alternatives, one-sided formulations). However, because the stem does not specify the test, multiple options remain equally plausible, weakening distractor effectiveness.

The question avoids answer leakage and is not tautological, but it is fundamentally underdetermined: the stem gives only the Bernoulli model, not the actual hypotheses. As a result, it does not genuinely test reasoning, and the keyed answer is unsupported by the provided information.


\begin{coro}\label{cor:sampling-numbers-H}
Let $(D,\mu)$ be a measure space
and $H$ a reproducing kernel Hilbert space
satisfying \eqref{cond:continuous}, \eqref{cond:injective} and \eqref{eq:finite-trace}.
Then, for all $m\in \N$ and $n\geq m$,
\[
g_n(B_H,L_2)
\,\leq\, \left(1+\frac{1}{1-r}\right)
\left(\sigma_m + \sqrt{\frac{1}{n}\sum_{k> m} \sigma_k^2} \right),
\]
where $r=\sqrt\frac{m-1}{n}$ and $\sigma_0 \geq \sigma_1 \geq \hdots$ are the singular values
of the embedding $H\hookrightarrow L_2$.
\end{coro}

\begin{prop}
\label{coro:intro-equal}
Let $(D,\mu)$ be a probability space, and
$V_m\subset L_2(D,\mu)\cap B(D)$ with $\dim(V_m)=m$.
For $n\geq m$, there exist points $x_1,\hdots,x_{2n}\in D$ such that,
for all $f\in V_m$ and $2\leq p\leq\infty$,
\[
\left(1-\sqrt{\frac{m}{n}}\right)
\cdot \|f\|_p
\;\leq\;
\,m^{\frac12-\frac1p}\cdot
\sqrt{\frac{1}{n}\sum_{i=1}^{2n} |f(x_i)|^2}.
\]
In the case $p\in\{2,\infty\}$, we can take $n$ points instead of $2n$ points.\\
In the case $n=m$, the factor $1-\sqrt{\frac{m}{n}}$ can be replaced by $\frac{1}{2m}$.
\end{prop}

\begin{prop}
\label{rk:weight_control}
Let the assumptions of Corollary~\ref{coro:main} hold.
If $(D,\mu)$ is a probability space and $\int f\,d\mu=0$ for all $f\in H$,
then for $n\geq m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$
such that \eqref{eq-low-coro-main} holds, \eqref{eq-up-coro-main} holds with $M$ instead of $M-1$, and
\[
\sum_{i=1}^n w_i \leq \left(1+\sqrt\frac{M}{n}\right)^2.
\]
\end{prop}

\begin{coro}
\label{coro:noisy-recovery}
Assume that $(D,\mu)$ is a probability space, that $H$ satisfies conditions \eqref{cond:continuous}, \eqref{cond:injective} and \eqref{eq:finite-trace}, and that $V_m \subset H$ has dimension $m$.
Moreover, assume that
$V_m$ contains the Riesz representer of the integral, i.e.,
\[
h=\int K(x,\cdot)\,d\mu(x) \,\in V_m.
\]
Then, for $n \ge m$,
there are points
$x_1,\hdots,x_n \in D$
and weights $w_1,\hdots,w_n>0$
such that, for any $f\in H$ and $e\in \C^n$, the least squares approximation \eqref{def:wls} on $V_m$ with noisy measurements $y_i=f(x_i)+e_i$ satisfies
\begin{equation}
\label{eq-prop-noisy}
\big\|f- \tilde f\big\|_2
\,\leq\,
\frac{1+\tilde s}{1-r}\, \Big(2\,\sqrt{\lambda_m} \,\big\|f-P_m f\big\|_H
+\,\Vert e \Vert_{\infty}\Big),
\end{equation}
with $\lambda_m$, $K_m$, $P_m$ and $r$ as in Corollary~\ref{cor:recovery},
and $\tilde s=\sqrt{\frac{\Tr(K_m)}{n\,\lambda_m}}$.
\end{coro}

\begin{theorem}
\label{thm:main}
Let $(D,\mu)$ be a measure space, and let $a=(a_1,\hdots,a_m)^\top$
and $b= (b_k)_{k\in \mathbb I}^\top$ be families of square-integrable functions on $D$,
where $\mathbb I$ is at most countable.
Assume that $I:=\int_D a(x)a(x)^*d\mu(x)$ and $J:=\int_D b(x)b(x)^*d\mu(x)$ are positive definite, and that $J$
has finite trace and effective dimension
\[
\Tr(J) \,=\, \sum_{k\in \mathbb I} \|b_k\|_{L_2(D,\mu)}^2 \,<\,\infty
\quad\text{and}\quad M:=\frac{\Tr(J)}{\lambda_{\max}(J)}.
\]
Then, for any
$n\geq m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\[
\lambda_{\min}\left( \sum_{i=1}^{n} w_i\,
a(x_i)a(x_i)^* \right) \geq
\left(1-\sqrt{\frac{m-1}{n}}\right)^2
\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\left(\sum_{i=1}^{n} w_i\, b(x_i)b(x_i)^* \right) \leq \left(1+\sqrt{\frac{M-1}{n}}\right)^2
\lambda_{\max}(J).
\]
\end{theorem}

\begin{example}
\label{ex:mixed-smoothness}
    An example, where Theorem~\ref{thm:constructive} could be applied
    is the approximation on Sobolev spaces $H^\alpha_{\rm mix}$
    of $d$-variate periodic functions with mixed smoothness $\alpha>1/2$.
    These can be defined by
    \[
     H^\alpha_{\rm mix} =
     \left\{ f \in C([0,1]^d) \colon \Vert f \Vert_{\alpha}^2 := \sum_{ k \in \Z^d} w_{ k}^{2\alpha} |\hat{f}( k)|^2 < \infty \right\},
    \]
    where $\hat{f}( k)$ are the usual Fourier coefficients and $w_{ k} = \prod_{i=1}^d \max\{1,|k_i|\}$.
    For integer $\alpha$, they can also be defined as spaces of functions possessing (weak) derivatives
    in $L_2$ up to mixed order $\alpha$,
    see, e.g., \cite[Sec.\,2.1]{KSU} for a few details,
    and \cite{DTU} for a comprehensive treatment of dominating mixed smoothness.
    If we choose the functions $\eta_k$ as the trigonometric monomials
    with the frequencies $k \in \Z^d$ ordered according to their hyperbolic distance $w_{k}$ to the origin
    and $\alpha_0>1/2$,
    the algorithm from Theorem~\ref{thm:constructive} satisfies
    \begin{equation}\label{eq:order-mix}
     \Vert f - \tilde f \Vert_2 \,\lesssim\, m^{-\alpha}
     (\log m)^{\alpha(d-1)}
     \cdot \Vert f \Vert_{\alpha},
    \end{equation}
    for any smoothness $\alpha\geq \alpha_0$.

    We present this example since
    for the class $H^\alpha_{\rm mix}$,
    no (polynomial time) construction was previously known to give the optimal order~\eqref{eq:order-mix}.
    Only the existence of such algorithms was known due to~\cite{DKU}.
    Previous constructions that come close to the optimal order include sparse grids (Smolyak's algorithm), see~\cite{SU},
    random points~\cite{KU1},
    subsamples of random points~\cite{BSU}
    and generated sets (lattices with a non-integer generating vector), see~\cite{CNW}.
    \end{example}

\label{subsec:numerics}

We now discuss some numerical examples
for the sampling points and weights
from Theorem~\ref{thm:constructive}.
Figures~\ref{fig:trig-blocked}--\ref{fig:legendre-new} show th

\begin{prop}[\cite{KW}]\label{prop:KW}
Let $D$ be a set and $V_m$ be an $m$-dimen\-sion\-al subspace of~$B(D)$.
For every $\varepsilon>0$,
there exists a finitely supported measure $\mu_\eps$ on~$D$,
such that
\[
 \|f\|_{\infty} \,\leq\, \sqrt{m+\varepsilon} \cdot \Vert f \Vert_{L_2(D,\mu_\eps)}.
\]
\end{prop}

\begin{theorem}
\label{thm:constructive}
Let $(D,\mu)$ be a probability space, $(\eta_k)_{k\in\II}$ an $L_2$-orthonormal system and $(I_\ell)_{\ell\in\N}$ nested index sets with $|I_\ell|\le\ell$ that satisfy condition~\eqref{eq:assum-uniform-bound}, $m\geq 2$, $n=2 m$, $\alpha_0>\theta$ and $N=\lceil m^{\alpha_0/(\alpha_0-\theta)}\rceil$.
Assume that the constant function $1$ belongs to $\{\eta_k,\;k\in I_m\}$. Algorithm~\ref{alg:abstract-construction} can be implemented, with $a$ and $b$ defined in \eqref{eq:a-b-constructive}, in time
\[
O\big((c+Nn)n^2\mathcal G\big),
\]
where $c$ is the cost of drawing one point from the Christoffel density, and ${\mathcal G\in\N}$ is a shifted geometric random variable of parameter $1/2$. It yields points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$ which satisfy the following.
If $f\in L_2(D,\mu)$ with
\[
\|f-f_{I_\ell}\|_2\leq C_f\, \ell^{-\alpha}\log^\beta(\ell)
\]
for some $\alpha\geq \alpha_0$ and $\beta\geq 0$, and if
$f(x_i)=\lim_{\ell\to\infty}f_{I_\ell}(x_i)$ for $1\leq i\leq n$,
the weighted least-squares approximation \eqref{def:wls}
from these values satisfies
\[
\|f-\tilde f\|_2\leq C_f\,C\, m^{-\alpha}\log^\beta(m),
\]
where $C$ depends on $\theta$, $\alpha_0$, $\alpha$, $\beta$ and $C_\eta$.
\end{theorem}

\begin{coro}\label{cor:recovery-unif}
Let $(D,\mu)$ be a probability space, $V_m\subset L_2(D,\mu)\cap B(D)$ with $\dim(V_m)=m$,
and $2\le p\le\infty$.
For all $n> m$,
there are points $x_1,\hdots,x_{2n} \in D$
such that the plain least squares
approximation~\eqref{def:pls}
with $y_i = f(x_i) + e_i$
satisfies
\[
\big\|f- \tilde f\big\|_p
\,\leq\, \left(1 + \sqrt{2}\cdot\frac{m^{1/2-1/p}}{1-\sqrt{m/n}}\right) \Big( \min_{g\in V_m} \,\big\|f-g\big\|_\infty + \Vert e \Vert_\infty \Big)
\]
for all $f\in B(D)$ and any noise $e_1,\hdots,e_n \in \C$.
For $p\in\{2,\infty\}$, we can use $n$ instead of $2n$ points, remove the factor $\sqrt{2}$,
and it suffices to have $V_m \subset L_2$ or $V_m \subset B(D)$, respectively.
\end{coro}

\label{sec:intro}
In recent years, much
progress has been made in norm discretization and sampling recovery, first
by considering concentration inequalities for random points,
then by applying a sample sparsification using \cite{MSS} or \cite{BSS}.
This follows the trend observed in other fields, including graph sparsification, frame discretization, matrix
sketching, subspace selection, and more generally randomized numerical linear algebra.

Here, we prove a generalization of the result from~\cite{BSS} by Batson, Spielman, and Srivastava (BSS), and expand on its implications
for the problems of norm discretization and sampling recovery.
The original result from~\cite{BSS} can be stated as follows:

\begin{prop}[\cite{BSS}, Theorem 3.1]
\label{thm:BSS}
Let $D$ be a finite set and let $\mu$ be the uniform distribution on $D$. Let $a=(a_1,\hdots,a_m)^\top$
be an orthonormal family in $L_2(D,\mu)$.
Then, for any $n>m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\[
\lambda_{\min}\left( \sum_{i=1}^n w_i\, a(x_i)a(x_i)^* \right) \geq
\left(1-\sqrt{\frac{m}{n}}\right)^2
\]
and
\[
\lambda_{\max}\left(\sum_{i=1}^n w_ia(x_i)a(x_i)^* \right) \leq \left(1+\sqrt{\frac{m}{n}}\right)^2.
\]
\end{prop}

Here and in the following, we denote by
$\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ the smallest and largest eigenvalues of $A$.
The points and weights from Proposition~\ref{thm:BSS} can be constructed in polynomial time.

In \cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \cite{HGN+24,SS11,ST11}.
For our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\mu)$; by decoupling the lower and upper frame bounds with two different
families of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families.

The first two kinds of generalizations are already available in the literature. General probability spaces are treated, for instance, in \cite[Theorem~6.4]{DPSTT}.
The second kind of generalization was achieved in \cite{BDM14}.
The result of \cite{BDM14}, called dual set spectral sparsification, can be reformulated as follows.

\begin{prop}[\cite{BDM14}, Lemma 13]
\label{prop:BDM}
Let $D$ be a finite set and let $\mu$ be the uniform distribution on $D$. Let $a=(a_1,\hdots,a_m)^\top$ and $b=(b_1,\dots,b_M)^\top$
be orthonormal families in $L_2(D,\mu)$.
Then for any integer $n>m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\[
\lambda_{\min}\left( \sum_{i=1}^{n} w_i\,
a(x_i)a(x_i)^* \right) \geq
\left(1-\sqrt{\frac{m}{n}}\right)^2
\]
and
\[
\,
\lambda_{\max}\left(\sum_{i=1}^{n} w_i\, b(x_i)b(x_i)^* \right) \leq \left(1+\sqrt{\frac{M}{n}}\right)^2.
\]
\end{prop}

Our main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the
upper frame bound.
\smallskip

\label{sec:intro}
In recent years, much
progress has been made in norm discretization and sampling recovery, first
by considering concentration inequalities for random points,
then by applying a sample sparsification using \cite{MSS} or \cite{BSS}.
This follows the trend observed in other fields, including graph sparsification, frame discretization, matrix
sketching, subspace selection, and more generally randomized numerical linear algebra.

\begin{prop}[\cite{BSS}, Theorem 3.1]
\label{thm:BSS}
Let $D$ be a finite set and let $\mu$ be the uniform distribution on $D$. Let $a=(a_1,\hdots,a_m)^\top$
be an orthonormal family in $L_2(D,\mu)$.
Then, for any $n>m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\[
\lambda_{\min}\left( \sum_{i=1}^n w_i\, a(x_i)a(x_i)^* \right) \geq
\left(1-\sqrt{\frac{m}{n}}\right)^2
\]
and
\[
\lambda_{\max}\left(\sum_{i=1}^n w_ia(x_i)a(x_i)^* \right) \leq \left(1+\sqrt{\frac{m}{n}}\right)^2.
\]
\end{prop}

Here and in the following, we denote by
$\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ the smallest and largest eigenvalues of $A$.
The points and weights from Proposition~\ref{thm:BSS} can be constructed in polynomial time.

In \cite{BSS}, this theorem is applied to the problem of graph sparsification, see also \cite{HGN+24,SS11,ST11}.
For our applications, we want to generalize the result in three ways: by allowing general measure spaces $(D,\mu)$; by decoupling the lower and upper frame bounds with two different
families of $L_2$-functions; and by making the upper frame bound independent of the dimension, thus allowing to consider very large and even infinite-dimensional families.

\begin{prop}[\cite{BDM14}, Lemma 13]
\label{prop:BDM}
Let $D$ be a finite set and let $\mu$ be the uniform distribution on $D$. Let $a=(a_1,\hdots,a_m)^\top$ and $b=(b_1,\dots,b_M)^\top$
be orthonormal families in $L_2(D,\mu)$.
Then for any integer $n>m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\[
\lambda_{\min}\left( \sum_{i=1}^{n} w_i\,
a(x_i)a(x_i)^* \right) \geq
\left(1-\sqrt{\frac{m}{n}}\right)^2
\]
and
\[
\,
\lambda_{\max}\left(\sum_{i=1}^{n} w_i\, b(x_i)b(x_i)^* \right) \leq \left(1+\sqrt{\frac{M}{n}}\right)^2.
\]
\end{prop}

Our main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the
upper frame bound.
\smallskip

\begin{prop}[\cite{BDM14}, Lemma 13]
\label{prop:BDM}
Let $D$ be a finite set and let $\mu$ be the uniform distribution on $D$. Let $a=(a_1,\hdots,a_m)^\top$ and $b=(b_1,\dots,b_M)^\top$
be orthonormal families in $L_2(D,\mu)$.
Then for any integer $n>m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\[
\lambda_{\min}\left( \sum_{i=1}^{n} w_i\,
a(x_i)a(x_i)^* \right) \geq
\left(1-\sqrt{\frac{m}{n}}\right)^2
\]
and
\[
\,
\lambda_{\max}\left(\sum_{i=1}^{n} w_i\, b(x_i)b(x_i)^* \right) \leq \left(1+\sqrt{\frac{M}{n}}\right)^2.
\]
\end{prop}

Our main contribution, stated below, is the third kind of generalization. Namely, we replace the dimension $M$ by an effective dimension in the
upper frame bound.
\smallskip

\smallskip

\begin{coro} \label{coro:main}
Let $(D,\mu)$ be a measure space,
$V_m\subset L_2(D,\mu)$ be an $m$-dimensional space of functions and let $H$ be
a reproducing kernel Hilbert space satisfying conditions \eqref{cond:continuous}, \eqref{cond:injective} and \eqref{eq:finite-trace}.
For $n\geq m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\begin{equation}
\label{eq-low-coro-main}
\left(1-\sqrt{\frac{m-1}{n}}\right)\,\|f\|_2
\,\leq\,
\sqrt{\sum_{i=1}^n w_i|f(x_i)|^2}
\qquad \text{for all } f \in V_m
\end{equation}
and
\begin{equation}
\label{eq-up-coro-main}
\sqrt{\sum_{i=1}^n w_i|g(x_i)|^2}
\,\leq\,
\left(1 +\sqrt{\frac{M-1}{n}} \right)\, \sqrt\lambda\,
\left\| g \right\|_H
\qquad \text{for all } g \in H,
\end{equation}
where $M=\Tr(K)/\lambda$ is the ``effective dimension'' of $H$ in $L_2(D,\mu)$.
\end{coro}

\begin{coro}\label{cor:recovery}
Let $(D,\mu)$ be a measure space.
Let $H$ be a reproducing kernel Hilbert space satisfying conditions \eqref{cond:continuous}, \eqref{cond:injective} and \eqref{eq:finite-trace}.
Let $V_m \subset H$ be an $m$-dimensional subspace and
let $n\geq m$.
Then, there are points $x_1,\hdots,x_n \in D$
and weights $w_1,\hdots,w_n>0$
such that, for any $f\in H$, the weighted least squares approximation~\eqref{def:wls} with $y_i=f(x_i)$ satisfies
\[
\big\|f- \tilde f\big\|_2
\,\leq\,
\left(1+\frac{1+s}{1-r}\right)\sqrt{\lambda_m} \,\big\|f-P_m f\big\|_H,
\]
where
\[
\lambda_m:=\sup_{f \in H_m} \frac{\Vert f \Vert_2^2}{\Vert f \Vert_{H}^2},
\qquad
r:=\sqrt\frac{m-1}{n}
\quad\text{and}\quad
s:=\sqrt{\frac{\Tr(K_m)-\lambda_m}{n\,\lambda_m}}.
\]
Here, $H_m$ is the $H$-orthogonal complement of $V_m$,
$P_m$ is the $H$-orthogonal projection onto $V_m$,
and $K_m$ is the reproducing kernel of $(H_m,\|\cdot\|_H)$.
\end{coro}

\begin{lemma}
\label{lem:accept_proba}
If $x$ is drawn according to the Christoffel measure $\rho$ and the increment $\delta=\frac{1-r}{n}$ is replaced by
\[
\delta_\epsilon=\frac{1-r-\epsilon}{n}
\]
for some parameter $\epsilon < (1-r)^2$ in the computation of $L_A^\delta$ and the updates $A\leftarrow A'$ (but not in the initialization $A_0$),
then the acceptance probability satisfies
\[
\mathbb P\left(L_A^{\delta_\eps}(a(x)) \ge U_B^\zeta(b(x))\right) \ge \frac{\epsilon}{m}.
\]
In addition, any points $x_1,\dots x_n$ returned by our algorithm for the modified value of $\delta$ satisfy the upper bound \eqref{eq:UB} while the lower bound \eqref{eq:LB} becomes
\[
\lambda_{\min}\left(\sum_{i=1}^{n} w_ia(x_i)a(x_i)^*\right)
\geq \Big(n\delta_\epsilon+\frac{\delta}{r}-\frac{m\delta}{r}\,\Big)= (1-r)^2-\epsilon.
\]
\end{lemma}
In the sequel, we will take $\epsilon=\frac{(1-r)^2}{2}$, which changes \eqref{eq:LB} by a factor 2.
\begin{proof}
We start with $A \in \mathcal P_m$ and $B\in \mathcal P$ such that \eqref{ineq-lemboth} holds for the original $\delta=\frac{1-r}{n}$.
Let
\[
\Delta(x):= \frac{L_A^{\delta_\epsilon}(a(x))- U_B^\zeta(b(x))}{\sum_{k=1}^m|a_k(x)|^2}.
\]
We now consider a random variable $x\sim \rho$.
According to Lemma~\ref{lemma:pos_measure},
the expectation of $\Delta(x)$ is
\begin{align*}
\E(\Delta(x))
=\int_D \frac{L_A^{\delta_\epsilon}(a(x))-U_B^\zeta(b(x))}m \, d\mu(x)
\geq \frac{1}{m}\left(\frac{1}{\delta_\epsilon}-\Phi(A)- \frac{1}{\zeta}-\Psi(B)\right).
\end{align*}
Thanks to condition \eqref{ineq-lemboth},
we get
\[
\E(\Delta(x))
\geq \frac{1}{m}\left(\frac{1}{\delta_\epsilon}-\frac{1}{\delta}\right)=\frac{\epsilon}{m \,\delta_\epsilon(1-r)}.
\]
On the other hand, $\Delta(x)$ is almost surely bounded by
\begin{align*}
\Delta(x)
\leq \sup_{\substack{a\in\C^m
}} \frac{L_A^{\delta_\eps}(a)}{\sum_{k=1}^m|a_k|^2}
\leq \frac{\lambda_{\max}(Z^2)}{\Tr(Z)-\Tr(Y)}\leq \frac{\Tr(Z^2)}{\delta_\epsilon\Tr(YZ)}\leq \frac{1}{\delta_\epsilon(1-r)},
\end{align*}
where $Y$ and $Z$ are from Section~\ref{sec:main-proof} with $\delta_\eps$ in place of $\delta$. Here,
the third inequality follows from $Z-Y=\delta_\epsilon YZ$
and the last inequality comes from the fact that
\[
\Tr(Z^2)-\Tr(YZ)=\delta_\epsilon \Tr(Y Z^2)\leq \delta_\epsilon\Tr(Y)\Tr(Z^2)\leq r\Tr(Z^2),
\]
by a Cauchy-Schwarz inequality~\eqref{property:CS} between $Y^{1/2}$ and $Y^{1/2}Z$ and
since $\Tr(Y)=\Phi(A)\leq r/\delta_\epsilon$. In the end, we obtain an acceptance probability
\[
\mathbb P(\Delta(x)\geq 0)
\geq \frac{\int_D \Delta\, d\rho}{\sup_{x\in D} \Delta(x)}\geq \frac{\epsilon}m.
\]
Note that, by Lemma~\ref{lemboth}, choosing $x_i$ with $\Delta(x_i) \ge 0$ and $w_i$ accordingly, we obtain that the potentials do not increase
when switching to the updates
$A':=A-\delta_\eps I+wa(x_i)a(x_i)^*$ and
$ B':=B+\zeta J-wb(x_i)b(x_i)^*$ and
so
condition~\eqref{ineq-lemboth} remains true with the original $\delta$ also for the updated operators.
\end{proof}

\begin{example}
\label{ex:christoffel}
For each $m\in\N$, consider $D=[0,m]$ with the Lebesgue measure $\mu$
and define a RKHS $H$ on $D$ by its kernel $K(x,y)=\sum_{k\in \II} \sigma_k^2 b_k(x) b_k(y)$, where the functions
\[
b_k := \left\{\begin{array}{cl}
\mathbbm 1_{[k-1,k]} &\text{ for } 1\leq k\leq m,\\
\cos(2\pi (k-m)x) \mathbbm 1_{[0,1]} &\text{ for } k>m,\\
\sin(2\pi (m-k)x) \mathbbm 1_{[0,1]} &\text{ for } k<-m,\\
\end{array}\right.
\]
are orthonormal in $L_2(D)$,
and $\sigma_k:=\frac1{|k|}$ for $k\in\II:=\Z\setminus\{-m,\dots,0\}$. Let also $V_m=\Span\{b_k \colon 1\leq k\leq m\}$. From~\eqref{eq:n-is-2m}, we get
\[
 g_{2m}^{\rm lin}(B_H,L_2) \,\leq\, \frac{10}m.
\]
On the other hand,
let $A_n$ be an approximation that uses information from at
most $n$ samples $x_1,\hdots,x_n$.
Let $\{y_1,\hdots,y_{n'}\} = \{x_1,\hdots,x_n\} \cap [0,1]$
be the samples in $[0,1]$. Observing that $A_n(-f)=A_n(f)$ if $f(x_1)=\dots f(x_n)=0$,
\[
 \sup_{\Vert f \Vert_H \leq 1} \Vert f - A_n(f) \Vert_{L_2([0,m])}
 \geq \sup_{\substack{\Vert f \Vert_H \leq 1 \\ f(x_i)=0}} \Vert f \Vert_{L_2([0,m])}
 \geq \sup_{\substack{\Vert f \Vert_H \leq 1 \\ f(y_i)=0}} \Vert f \Vert_{L_2([0,1])}.
 \]
As $\|f\|_{L_2([0,1])}\geq \big|\int_0^1f\big|$, the lower bound from \cite[Corollary~2]{KV} with $\lambda_0=1$ and $\lambda_k=\sigma_{|k|+m}^2$ for $k\neq 0$ gives
 \[
\sup_{\substack{\Vert f \Vert_H \leq 1 \\ f(y_i)=0}} \Vert f \Vert_{L_2([0,1])}
 \geq \sqrt{\min\bigg\{\frac{\lambda_0}2, \ \frac{1}{8n'} \sum_{k> 4n'} \lambda_k\bigg\}}
 \geq \frac{1}{8\max(n',\sqrt{mn'})}.
\]
As a consequence, the approximation $A_n$ can only attain an optimal error if $n' \gtrsim m$.
As the Christoffel density of $V_m$ is uniform on $[0,m]$,
if $x_1,\hdots,x_n$ are drawn i.i.d. from that density,
we must have $n \gtrsim m^2$ with high probability. In summary, any
algorithm achieving an error of optimal order
must use $\Omega(m^2)$ Christoffel samples.
\end{example}

\smallskip

One can retrieve Proposition~\ref{prop:BDM} from Theorem~\ref{thm:main} by taking $\mathbb I=\{1,\dots,M\}$ and assuming that $a$ and $b$ are orthonormal families. Indeed, in that case, we see that $I$ and $J$ are the $m\times m$ and $M\times M$ identity matrices. In turn, Proposition~\ref{prop:BDM} implies Proposition~\ref{thm:BSS} by taking $M=m$ and $b=a$.

\begin{remark}
The factor $m-1$ instead of $m$ in the lower bound is a minor improvement
already appearing in \cite{ChkifaDolbeault24}. It allows to treat the case $n=m$, with the bound
\[
1-\sqrt{\frac{m-1}{m}}>\frac{1}{2m}.
\]
In the same way, the factor $M-1$ instead of $M$
in the upper bound is most interesting when $J$ has rank one, where we recover \cite[Lemma~14]{BDM14}.
This case will be used in Corollary~\ref{coro:intro-equal} to obtain discretizations with equal weights. We will also leverage these improvements in Propositions~\ref{rk:weight_control} and Corollary~\ref{coro:noisy-recovery}, when considering families $a$ and $b$ containing one more function.
\end{remark}

\begin{remark}
Throughout the article, $L_2(D,\mu)$ and its subspaces are assumed to consist of complex-valued functions, not equivalence classes, which is needed for discretization.
Moreover, it is possible to extend our results to Hilbert-valued functions, by combining them with \cite[Theorem 2.1]{BartelDung24}.
\end{remark}

The proof of Theorem~\ref{thm:main}, see Section~\ref{sec:proof},
builds upon the original proof of Proposition~\ref{thm:BSS},
and the points and weights can be constructed in a quite similar way, see Algorithm~\ref{alg:abstract-construction} in Section~\ref{sec:implementation}.
Although the proof of Theorem~\ref{thm:main} is self-contained,
we recommend reading the proof of \cite[Theorem~3.1]{BSS} first, since it comes with a very nice physical intuition.

Before we come to the proof, we first discuss several applications of this linear algebra result.
In Section~\ref{sec:discretization}, we discuss its implications to the problem of norm discretization.
In Section~\ref{sec:sampling}, we use these discretization results to obtain results on the problem of sampling recovery
and on the error of (weighted) least squares algorithms. The relation between these results is illustrated in Figure~\ref{fig:graph-results}.

\begin{figure}[H]
\label{fig:graph-results}
\centering
\begin{tikzcd}
\rm Prop.~\ref{thm:BSS}
&\rm Prop.~\ref{prop:BDM} \arrow[l]
&\rm Thm.~\ref{thm:main} \arrow[l] \arrow[ld] \arrow[rd]
&\rm Prop.~\ref{prop:KW} \arrow[d] \\
\rm Prop.~\ref{rk:weight_control} \arrow[d]
&\rm Cor.~\ref{coro:main} \arrow[l] \arrow[d] \arrow[rr]
& &\rm Prop.~\ref{coro:intro-equal} \arrow[d] \\
\rm Cor.~\ref{coro:noisy-recovery}
&\rm Cor.~\ref{cor:recovery} \arrow[l] \arrow[r]
&\rm Cor.~\ref{cor:sampling-numbers-H}
&\rm Cor.~\ref{cor:recovery-unif}
\end{tikzcd}
\caption{Summary of the results of Sections~\ref{sec:intro}, \ref{sec:discretization} and \ref{sec:sampling}, with arrows denoting implications. The first line contains linear algebra results, the second deals with norm discretization and the third with sampling recovery. The lower left block details the reproducing kernel Hilbert space (RKHS) setting, with the second column directly applying Thm.~\ref{thm:main}, while the first column adds a control on the weights, and the third looks at sampling numbers. Lastly, the fourth column is focused on the $L_p$-setting, in which all weights are equal.}
\end{figure}

In fact, it is quite interesting that many of the recent advances in
the areas of both norm discretization and sampling recovery
can be derived solely on the basis of Theorem~\ref{thm:main}.
Previous results used the non-constructive Kadison-Singer theorem from \cite{MSS},
see, e.g., \cite{CohenDolbeault,DKU,NSU},
which now becomes unnecessary.
This does not only simplify the proofs of those results
but also much improves the absolute constants.
Moreover, the least squares algorithms based on Theorem~\ref{thm:main}
are implementable for various examples where
an implementable construction of an optimal algorithm
has previously been unknown
(e.g., for function spaces of mixed smoothness),
see Theorem~\ref{thm:constructive} and Example~\ref{ex:mixed-smoothness}.
A discussion of the practical implementation, along with numerical illustrations, is provided in Sections~\ref{sec:implementation} and~\ref{subsec:numerics}.

The proof of Theorem~\ref{thm:main} (see Section~\ref{sec:proof}) is said to “builds upon the original proof of Proposition~\ref{thm:BSS},” and the “points and weights can be constructed in a quite similar way,” referring to Algorithm~\ref{alg:abstract-construction} in Section~\ref{sec:implementation}. Although the proof is “self-contained,” the authors recommend reading the proof of \cite[Theorem~3.1]{BSS} first “since it comes with a very nice physical intuition.”

The proof of the main theorem (see Section~\ref{sec:proof}) is said to “builds upon the original proof of \begin{prop}[\cite{BSS}, Theorem 3.1]
\label{thm:BSS}
Let $D$ be a finite set and let $\mu$ be the uniform distribution on $D$. Let $a=(a_1,\hdots,a_m)^\top$
be an orthonormal family in $L_2(D,\mu)$.
Then, for any $n>m$, there exist points $x_1,\dots,x_n\in D$ and weights ${w_1,\dots,w_n>0}$ such that
\[
\lambda_{\min}\left( \sum_{i=1}^n w_i\, a(x_i)a(x_i)^* \right) \geq
\left(1-\sqrt{\frac{m}{n}}\right)^2
\]
and
\[
\lambda_{\max}\left(\sum_{i=1}^n w_ia(x_i)a(x_i)^* \right) \leq \left(1+\sqrt{\frac{m}{n}}\right)^2.
\]
\end{prop},” and the “points and weights can be constructed in a quite similar way,” referring to Algorithm~\ref{alg:abstract-construction} in Section~\ref{sec:implementation}:
\[
\label{alg:abstract-construction}
\end{algorithm}

Regarding the first issue, one option is to find a finite subset of $D$ in which at least one point satisfies the condition of line~\ref{state:test}.
\]
Although the proof is “self-contained,” the authors recommend reading the proof of \cite{BSS}, Theorem~3.1 first “since it comes with a very nice physical intuition.”

Question:
Let $(D,\mu)$ be a measure space. Let $a=(a_1,\dots,a_m)^\top$ and $b=(b_k)_{k\in\mathbb I}^\top$ be families of square-integrable functions on $D$, where $\mathbb I$ is at most countable. Define
\[
I:=\int_D a(x)a(x)^*\,d\mu(x),\qquad J:=\int_D b(x)b(x)^*\,d\mu(x),
\]
and assume that both $I$ and $J$ are positive definite. Assume also that $J$ has finite trace
\[
\operatorname{Tr}(J)=\sum_{k\in\mathbb I}\|b_k\|_{L_2(D,\mu)}^2<\infty,
\]
and define its effective dimension by
\[
M:=\frac{\operatorname{Tr}(J)}{\lambda_{\max}(J)}.
\]
Here $\lambda_{\min}(\cdot)$ and $\lambda_{\max}(\cdot)$ denote the smallest and largest eigenvalues. For a given integer $n\ge m$, which quantitative estimate is valid?

There exist points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$ such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m-1}{n}}\right)^2\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\!\left(\sum_{i=1}^n w_i\,b(x_i)b(x_i)^*\right)
\le \left(1+\sqrt{\frac{M-1}{n}}\right)^2\lambda_{\max}(J).
\]

Choices:
B.
There exist points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$ such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m}{n}}\right)^2\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\!\left(\sum_{i=1}^n w_i\,b(x_i)b(x_i)^*\right)
\le \left(1+\sqrt{\frac{M}{n}}\right)^2\lambda_{\max}(J).
\]
C.
There exist points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$ such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m-1}{n}}\right)^2\lambda_{\min}(I).
\]
D.
There exist points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$, depending only on $n$, $m$, and $M$, such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m-1}{n}}\right)^2\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\!\left(\sum_{i=1}^n w_i\,b(x_i)b(x_i)^*\right)
\le \left(1+\sqrt{\frac{M-1}{n}}\right)^2\lambda_{\max}(J).
\]
E.
For every choice of points $x_1,\dots,x_n\in D$, there exist weights $w_1,\dots,w_n>0$ such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m-1}{n}}\right)^2\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\!\left(\sum_{i=1}^n w_i\,b(x_i)b(x_i)^*\right)
\le \left(1+\sqrt{\frac{M-1}{n}}\right)^2\lambda_{\max}(J).
\]

upper eigenvalue estimate for the $b$-family

dependence of selected points and weights on the actual families $a,b$ and operators $I,J$

existential choice of sample points produced by the construction

The stem does not explicitly reveal the correct option. It provides assumptions and definitions, but the exact two-sided eigenvalue conclusion in A is not stated in the prompt itself.

This is essentially a theorem-recall item: the question asks which existence statement holds under the listed assumptions, and choice A is the theorem’s conclusion almost verbatim rather than an application or derivation.

There is some reasoning required to distinguish the exact sharp form from nearby variants, especially because one distractor is a weaker true statement and others involve plausible parameter confusions. However, the main task is still recognition of the theorem statement rather than substantial generation.

The distractors are mathematically plausible and target realistic failure modes: replacing sharp constants by nearby ones, dropping one half of the conclusion, confusing effective dimension with trace, and mixing up λ_max(J) with Tr(J).

A technically well-constructed recall-style MCQ with strong distractors, but it is largely a direct restatement of a theorem rather than a genuinely generative reasoning task.


Question:
Let $(D,\mu)$ be a measure space. Let $a=(a_1,\dots,a_m)^\top$ and $b=(b_k)_{k\in\mathbb I}^\top$ be families of square-integrable functions on $D$, where $\mathbb I$ is at most countable. Define
\[
I:=\int_D a(x)a(x)^*\,d\mu(x),\qquad J:=\int_D b(x)b(x)^*\,d\mu(x),
\]
and assume that both $I$ and $J$ are positive definite. Assume also that $J$ has finite trace
\[
\operatorname{Tr}(J)=\sum_{k\in\mathbb I}\|b_k\|_{L_2(D,\mu)}^2<\infty,
\]
and define its effective dimension by
\[
M:=\frac{\operatorname{Tr}(J)}{\lambda_{\max}(J)}.
\]
Here $\lambda_{\min}(\cdot)$ and $\lambda_{\max}(\cdot)$ denote the smallest and largest eigenvalues. For a given integer $n\ge m$, which quantitative estimate is valid?

Choices:
B.
There exist points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$ such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m}{n}}\right)^2\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\!\left(\sum_{i=1}^n w_i\,b(x_i)b(x_i)^*\right)
\le \left(1+\sqrt{\frac{M}{n}}\right)^2\lambda_{\max}(J).
\]
C.
There exist points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$ such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m-1}{n}}\right)^2\lambda_{\min}(I).
\]
D.
There exist points $x_1,\dots,x_n\in D$ and weights $w_1,\dots,w_n>0$, depending only on $n$, $m$, and $M$, such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m-1}{n}}\right)^2\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\!\left(\sum_{i=1}^n w_i\,b(x_i)b(x_i)^*\right)
\le \left(1+\sqrt{\frac{M-1}{n}}\right)^2\lambda_{\max}(J).
\]
E.
For every choice of points $x_1,\dots,x_n\in D$, there exist weights $w_1,\dots,w_n>0$ such that
\[
\lambda_{\min}\!\left(\sum_{i=1}^n w_i\,a(x_i)a(x_i)^*\right)
\ge \left(1-\sqrt{\frac{m-1}{n}}\right)^2\lambda_{\min}(I)
\]
and
\[
\lambda_{\max}\!\left(\sum_{i=1}^n w_i\,b(x_i)b(x_i)^*\right)
\le \left(1+\sqrt{\frac{M-1}{n}}\right)^2\lambda_{\max}(J).
\]

The stem does not explicitly or implicitly reveal option A. It gives the hypotheses and asks for the valid estimate, but no wording singles out the exact constants or quantifiers in the correct choice.

This is essentially a theorem-recall item: the stem presents the full setup and asks which conclusion is valid. The correct option is basically the exact theorem statement rather than a conclusion derived from independent reasoning.

There is some reasoning pressure in checking sharp constants, completeness of the conclusion, and quantifier dependence across the options. However, the task is still mostly recognition of the exact theorem statement rather than substantive mathematical generation.

The distractors are strong: B alters sharp boundary terms, C gives a weaker partial conclusion, D mishandles dependence/quantifiers, and E improperly strengthens the existential claim to a universal one. These are plausible and mathematically meaningful failure modes.

A solid recall/discrimination MCQ with strong distractors and no answer leakage, but it is largely tautological because it mainly tests recognition of the theorem’s exact statement rather than deeper generative reasoning.