\label{1bdry C^s}
    Suppose $\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
    \begin{equation}
        \begin{cases}
            \mbox{$(-\triangle)^s$} u = g,&\text{in } \Omega\cap B_4,\\
            u=0, &\text{in } B_4\backslash \Omega.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
    \[
    \lVertu\rVert_{C^s(\Omega\cap B_{1/2})}\le C(\lVertg\rVert _{L^\infty(\Omega\cap B_4)} + \lVertu\rVert _{L^\infty(\Omega\cap B_4)} ),
    \]
    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.

\begin{theorem}\label{1bdry C^s}
    Suppose $\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
    \begin{equation}
        \begin{cases}
            \flap u = g,&\text{in } \Omega\cap B_4,\\
            u=0, &\text{in } B_4\backslash \Omega.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
    \[
    \norm{u}_{C^s(\Omega\cap B_{1/2})}\le C(\norm{g} _{L^\infty(\Omega\cap B_4)} + \norm{u} _{L^\infty(\Omega\cap B_4)} ),
    \]
    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.
\end{theorem}

\begin{theorem}\label{conv}
    Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
    converges in $C^{2s+\beta}_{loc}(\Omega)$ to a function $u\in \mathcal{L}_{2s}$ , and $\{\flap u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
    \begin{equation}
        \lim _{i\to\infty}\flap u_i(x) = \flap u(x) -b.
    \end{equation}
\end{theorem}

In this paper, we consider the following fractional problem in  unbounded domain $\Omega$
 \begin{equation}\label{main1}
        \begin{cases}
            \mbox{$(-\triangle)^s$} u(x) = g(x),&\text{in } \Omega,\\
            u(x)=0, &\text{in }  \Omega^c.\\
        \end{cases}
    \end{equation}
First, we derive a boundary regularity estimate for  solutions
in terms of only the local $L^{\infty}$ norm of $u$ and $g$.
Then as an important application, we obtain  a priori estimates for solutions to nonlinear equations on unbounded domains with  boundaries \begin{equation}\label{main2}
\begin{cases}
    \mbox{$(-\triangle)^s$} u (x) = f(x,u)  , & x\in \Omega,\\
    u (x)  = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}
Here, $(-\Delta)^s$  denotes the fractional Laplacian,  defined as
\begin{eqnarray}\label{eq1-1}
\begin{aligned}
(-\Delta)^s u(x)&=C_{n, s}P.V. \int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\
&=C_{n, s}\lim_{\varepsilon \to 0}\int_{\mathbb{R}^n\backslash B_\varepsilon(0)}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,
\end{aligned}
\end{eqnarray}
where $P.V.$
stands for the Cauchy principal value, and $C_{n, s}$ is a
dimensional constant.
In order that the integral on the right hand side of \eqref{eq1-1} is well defined in the classical sense, we require that $u\in C_{loc}^{1, 1}(\Omega)\cap \mathcal{L}_{2s}$, where
$$
\mathcal{L}_{2s}=\left\{u\in L_{loc}^1(\mathbb{R}^n) \,\Big| \int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx<\infty \right\}
$$
endowed with norm

\[\lVertu\rVert_{{\mathcal{L}}_{2s}}:=\int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx.\]

The fractional Laplacian is a nonlocal integro-differential operator that effectively models memory effects and long-range diffusion phenomena (Chaves \cite{Chave1998}; Metzler and Klafter \cite{MK2000}). Specific applications include random walks with memory, Lévy flights \cite{MS1984, SZK1993, N1986}, and kinetic theories of systems exhibiting chaotic \cite{SZ1997, Z2002} and pseudo-chaotic \cite{ZE2001} dynamics. Beyond these, the operator finds significant applications across diverse domains including probability theory and finance, image processing and computer vision, mathematical biology and ecology, as well as geophysics and climate modeling.

 Nonlocal operators, particularly the fractional Laplacian, have attracted considerable attention in the mathematical community owing to their broad applications. For foundational developments in this field, we refer to the seminal work of Caffarelli and Silvestre \cite{CS07}, along with subsequent contributions by  Stinga \cite{Stinga2019}, Dipierro et al. \cite{Dipierro2019}, and the comprehensive monograph by Bucur and Valdinoci \cite{BV2016}. Regarding qualitative properties of solutions, extensive results can be found in \cite{CDQ,CDG,CG,CGLO,CHL17,DQ, FL,FLS}, with specific advances in a priori estimates documented in \cite{CGL,Chen2016directblowup,ouyang2023}.

    The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \cite{Silvestre2007Regularity}, who established Schauder and H\"older estimates for entire solutions. Then the interior type estimates are studied in \cite{Chen2010frac}.  Later, Li and Wu \cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.
    For boundary regularity, Ros-Oton and Serra \cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.
    The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \cite{CS2016} and Stinga-Torrea \cite{ST2010} provided additional important regularity criteria.
    For more results concerning the nonlocal elliptic equations, please see \cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.

In particular, in \cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\"older and Schauder regularity estimates for fractional equation \eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \eqref{main2} when $\Omega=\mathbb{R}^n$, an  unbounded domain {\em without boundary}.

Then for general unbounded domains  with  boundaries, can one derive a priori estimates for the solutions?

This requires boundary H\"older estimates in terms of only the local $L^{\infty}$ norm of the solutions.

However, in the existing results, such as the ones in \cite{BDGQ2018} and \cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\"older continuity of the solutions for \eqref{main1} when $\Omega=\mathbb{R}^n_+$, the authors  require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \cite{Rosoton2014Dirichlet}, in which  an optimal boundary H\"older estimate in terms of $\lVertg\rVert_{L^\infty(\Omega)}$ is established for  a {\em bounded domian} $\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution  to show that the global $L^{\infty}$ norm of $u$ is bounded by $\lVertg\rVert_{L^\infty(\Omega)}$. When $\Omega$ is an {\em unbounded domain}, it is evident   from their proof that  in order to obtain local boundary H\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \cite{BDGQ2018} and \cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \cite{DSV17} obtain $C^{\alpha}$ boundary regularity for some $0<\alpha<s$ under the condition that the solution $u$ of \eqref{main1} be globally bounded.

With these existing boundary  regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.

One of our main result is to establish the local version of the boundary regularity for nonnegative solutions,  in which  we use {\em only the local $L^{\infty}$  norm of $u$ instead of the global ones} to control its local H\"older norm up to the boundary.

In this paper, we consider the following fractional problem in unbounded domain $\Omega$
 \begin{equation}\label{main1}
 \begin{cases}
 \mbox{$(-\triangle)^s$} u(x) = g(x),&\text{in } \Omega,\\
 u(x)=0, &\text{in } \Omega^c.\\
 \end{cases}
 \end{equation}
First, we derive a boundary regularity estimate for solutions
in terms of only the local $L^{\infty}$ norm of $u$ and $g$.
Then as an important application, we obtain a priori estimates for solutions to nonlinear equations on unbounded domains with boundaries \begin{equation}\label{main2}
\begin{cases}
 \mbox{$(-\triangle)^s$} u (x) = f(x,u) , & x\in \Omega,\\
 u (x) = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}
Here, $(-\Delta)^s$ denotes the fractional Laplacian, defined as
\begin{eqnarray}\label{eq1-1}
\begin{aligned}
(-\Delta)^s u(x)&=C_{n, s}P.V. \int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\
&=C_{n, s}\lim_{\varepsilon \to 0}\int_{\mathbb{R}^n\backslash B_\varepsilon(0)}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,
\end{aligned}
\end{eqnarray}
where $P.V.$
stands for the Cauchy principal value, and $C_{n, s}$ is a
dimensional constant.
In order that the integral on the right hand side of \eqref{eq1-1} is well defined in the classical sense, we require that $u\in C_{loc}^{1, 1}(\Omega)\cap \mathcal{L}_{2s}$, where
$$
\mathcal{L}_{2s}=\left\{u\in L_{loc}^1(\mathbb{R}^n) \,\Big| \int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx<\infty \right\}
$$
endowed with norm

The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \cite{Silvestre2007Regularity}, who established Schauder and H\"older estimates for entire solutions. Then the interior type estimates are studied in \cite{Chen2010frac}. Later, Li and Wu \cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.
 For boundary regularity, Ros-Oton and Serra \cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.
 The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \cite{CS2016} and Stinga-Torrea \cite{ST2010} provided additional important regularity criteria.
 For more results concerning the nonlocal elliptic equations, please see \cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.

In particular, in \cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\"older and Schauder regularity estimates for fractional equation \eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \eqref{main2} when $\Omega=\mathbb{R}^n$, an unbounded domain {\em without boundary}.

However, in the existing results, such as the ones in \cite{BDGQ2018} and \cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\"older continuity of the solutions for \eqref{main1} when $\Omega=\mathbb{R}^n_+$, the authors require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \cite{Rosoton2014Dirichlet}, in which an optimal boundary H\"older estimate in terms of $\lVertg\rVert_{L^\infty(\Omega)}$ is established for a {\em bounded domian} $\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution to show that the global $L^{\infty}$ norm of $u$ is bounded by $\lVertg\rVert_{L^\infty(\Omega)}$. When $\Omega$ is an {\em unbounded domain}, it is evident from their proof that in order to obtain local boundary H\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \cite{BDGQ2018} and \cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \cite{DSV17} obtain $C^{\alpha}$ boundary regularity for some $0<\alpha<s$ under the condition that the solution $u$ of \eqref{main1} be globally bounded.

With these existing boundary regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.

One of our main result is to establish the local version of the boundary regularity for nonnegative solutions, in which we use {\em only the local $L^{\infty}$ norm of $u$ instead of the global ones} to control its local H\"older norm up to the boundary.

\begin{equation}\label{main1}
        \begin{cases}
            \flap u(x) = g(x),&\text{in } \Omega,\\
            u(x)=0, &\text{in }  \Omega^c.\\
        \end{cases}
    \end{equation}

\begin{equation}\label{main2}
\begin{cases}
    \flap u (x) = f(x,u)  , & x\in \Omega,\\
    u (x)  = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}

As an important application, we establish a priori estimate for the solutions to \eqref{main2}. We assume that $\Omega \subset \R ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:
\begin{itemize}
 \item $f(x,t):\Omega \times [0,\infty)\to\R$ is uniformly H\"older continuous with respect to $x$ and continuous with respect to $t$.
\end{itemize}
\begin{theorem}\label{1A1}
Assume $1<p<\frac{n+2s}{n-2s}$, $f$ satisfies
\begin{equation}\label{growth f}
 f(x,t)\le C_0(1+t^p)
\end{equation}
uniformly for all $x$ in $\Omega$, and
\begin{equation}\label{lim f}
 \lim _{t\to\infty}\frac{f(x,t)}{t^p} = K(x),
\end{equation}
 where $K(x)\in (0,\infty)$ is uniformly continuous and $K(\infty):=\lim _{|x|\to\infty }K(x) \in (0,\infty)$.
Then there exists a constant $C$, such that
\begin{equation}\label{bd1}
u(x)\le C,\quad\forall \, x\in \Omega
\end{equation}
holds for all nonnegative solutions $u$ of \eqref{main2}.
\end{theorem}

To begin with, let us recall a classical result in
\cite{Rosoton2014Dirichlet}, in which the authors establish an optimal boundary regularity for the fractional equation with zero exterior Dirichlet condition.
\begin{proposition}[\cite{Rosoton2014Dirichlet}]\label{otonserra}
 Suppose $\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\in L^\infty(\Omega)$, $s\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:
 \begin{equation}\label{eq1}
 \begin{cases}
 \flap u = g,&\text{in } \Omega,\\
 u=0, &\text{in } \Omega ^c.\\
 \end{cases}
 \end{equation}
 Then $u\in C^{s}(\R^n)$ and
\begin{equation}\label{hd}
 \norm{u}_{C^s(\R^n)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation}
where $C=C(n,s)$ is a positive constant.
\end{proposition}
Notice that here $\Omega$ is a {\em bounded domian}. In this situation, a supersolution and a subsolution can be constructed to show that the global $L^{\infty}$ norm of $u$ is bounded by $\norm{g}_{L^\infty(\Omega)}$ as the following. Let
\[\psi(x):=(1-|x|^2)_+^s.\]
It is well known that
\[\flap \psi (x)=a>0.\]
Choose $R$ sufficiently large, so that $\Omega\subset B_R(0).$ Denote
\[\bar{u}(x):=\norm{g}_{L^{\infty}(\Omega)}\frac{R^{2s}}{a}\psi_R(x), ~\mbox{with}~ \psi_R(x):={\psi(\frac{x}{R})}.\]
Then it can be easily verified that $\bar{u}$ and $-\bar{u}$ are super-solution and sub-solution of \eqref{eq1}, respectively. Therefore,
\begin{equation*}
 \norm{u}_{L^{\infty}(\Omega)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation*}
From the above construction, one can see that in order to obtain \eqref{hd}, it is necessary that $\Omega$ be bounded.
When $\Omega$ is an {\em unbounded domain}, it is evident that the solution $u$ is required to be globally bounded. This requirement can not be fulfilled in the process of employing the blow-up and rescaling argument to obtain a priori estimates for solutions to a corresponding family of nonlinear fractional equations on {\em unbounded domains with boundaries}.

This motivates us to establish a local version of the boundary regularity, in which, instesd of global one, only a local ${L^{\infty}}$ norm of the solution is involved.
\begin{theorem}\label{bdry C^s}
 Suppose $\Omega$ is a unbounded domain with locally $C^{1,1}$ boundary, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
 \begin{equation}
 \begin{cases}
 \flap u = g,&\text{in } \Omega\cap B_4,\\
 u=0, &\text{in } B_4\backslash \Omega.\\
 \end{cases}
 \end{equation}
 Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
 \[
 \norm{u}_{C^s(\Omega\cap B_{1/2})}\le C(\norm{g} _{L^\infty(\Omega\cap B_4)} + \norm{u} _{L^\infty(\Omega\cap B_4)} ),
 \]
 where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.
\end{theorem}
The idea of the proof is that we divide the solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Theorem \ref{otonserra}. For the harmonic part, we firstly derived an $\alpha$ power order decay near $\partial\Omega$, here $\alpha$ is a certain positive number which may be less than $s$. Combining the decay estimate and the interior regularity result (Theorem \ref{Holder thm}), we derive the $C^\alpha$ boundary regularity. Finally we refine the boundary regularity by an iteration progress.

To continue estimate $\norm{u}_{C^{\alpha_0}(\Omega\cap B_{1/2})}$, we need the interior H\"older estimate from \cite{chen2025refinedregularitynonlocalelliptic}.
\begin{proposition}[Interior H\"older regularity]\label{Holder thm}
 Assume $0<s<1$, and $u\in C^{1,1}_{loc}(\R^n)\cap \mathcal{L}_{2s}$ is a nonnegative solution of
 \begin{equation}
 \flap u(x) = g(x),\quad x\in B_1
 \end{equation}
 where $g\in L^\infty(B_1)$. Then for any $0<\varepsilon <2s$,
 \begin{equation}
 \norm {u}_{C^{[2s-\varepsilon],\{2s-\varepsilon\}}(B_{1/2})}\le C(\norm{g}_{L^\infty(B_1)}+\norm{u}_{L^\infty(B_1)}).
 \end{equation}
\end{proposition}
Based on this, we arrive at
\begin{lemma}\label{global holder lemma}
 The decay estimate \eqref{decay estimate} and Proposition \ref{Holder thm} implies that
 \begin{equation}
 \norm{u} _{C^{\alpha _0}(\Omega\cap B_{1/2})}\le C(\norm{u}_{L^\infty(\Omega\cap B_4)}+\norm{g}_{L^\infty(\Omega\cap B_4)})
 \end{equation}
 where $C$ depends on $s,n$, and $\norm{\partial \Omega}_{C^{1,1}_{loc}}$, $\alpha_0:=\min\{s,1-s\}\in (0,1/2)$.
\end{lemma}

\section{Appendix}
Here is a generalization of \cite[Theorem 1.1]{Du2023blowup}Du-Jin-Xiong-Yang's convergence result.
\begin{theorem}
 Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
 converges to a function $u\in \mathcal{L}_{2s}$ in $C^{2s+\beta}_{loc}(\Omega)$, and $\{\flap u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
 \begin{equation}
 \lim _{i\to\infty}\flap u_i(x) = \flap u(x) -b.
 \end{equation}
\end{theorem}
\begin{proof}
 Fix any $x\in\Omega$, consider $R\gg|x|+1$.
 \begin{align*}
 \flap u(x)-\flap u_i(x) & = c_{n,s}\int _{B_R}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\ &+c_{n,s}\int _{B_R^c}\frac{(u-u_i)(x)-u(y)}{|x-y|^{n+2s}}dy\\
 & + c_{n,s}\int _{B_R^c}\frac{u_i(y)}{|x-y|^{n+2s}}dy\\
 & := A_i(x,R) + E_i(x,R) + F_i(x,R).
 \end{align*}
 For $E_i$, since $u\in \mathcal{L}_{2s}$,
 \begin{equation}
 \lim _{R\to\infty}\lim _{i\to\infty}E_i(x,R) = -c_{n,s}\lim _{R\to\infty}\int _{B_R^c}\frac{u(y)}{|x-y|^{n+2s}}dy =0.
 \end{equation}
 The consideration of $A_i$ is a little bit different with that in \cite{Du2023blowup}. In our cases, $B_R$ is not contained in $\Omega$ for sufficiently large $R$. Fix $\varepsilon >0$ such that $B_\varepsilon(x)\subset \Omega$. Then
 \begin{align*}
 A_i(x,R) & = c_{n,s}\int _{B_R\backslash B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\
 &+c_{n,s}\int _{B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy
 \end{align*}
 By dominated convergence theorem, the first term converges to 0 as $i\to\infty$. For the second term,
 \[
 \lim _{i\to\infty} c_{n,s}\int _{B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\le c_{n,s}\lim _{i\to\infty}[u-u_i]_{C^{2s+\beta}(B_\varepsilon(x))}\varepsilon ^\beta = 0.
 \]
 Hence $\lim _{i\to\infty}A_i(x,R)=0$. Therefore
 \begin{equation}
 \lim _{R\to\infty}\lim _{i\to\infty}A_i(x,R)=0.
 \end{equation}
 Then the same argument in \cite{Du2023blowup} implies $\lim _{i\to\infty}F_i(x,R)$ exists and
 \[
 \lim _{R\to\infty}\lim _{i\to\infty}F_i(x,R) =\lim _{R\to\infty}\lim _{i\to\infty}F_i(0,R) = :b\ge 0.
 \]
\end{proof}

The idea of the proof is that we divide a given solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Proposition \ref{otonserra}. For the harmonic part $h$, we rewrite it in terms of the Poisson representation formula in balls. Using this explicit expression, we  first carry out a detailed analysis to   derive an $\alpha$ power order decay near $\partial\Omega$,
\begin{equation*}
        |h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\operatorname{dist}(x,\partial \Omega)^{\alpha },
    \end{equation*}
    where $\alpha = \min\{s,1-s\}$. Combining this decay estimate and the interior regularity result (Theorem \ref{Holder thm}), we derive the $C^\alpha$ boundary regularity. Then by an iteration process, we increase the power $\alpha$ successively until it reaches the desired power $s$.

As an important application, we establish a priori estimate for the solutions  to \eqref{main2}.  We assume that $\Omega \subset \mathbb{R} ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:
\begin{itemize}
    \item $f(x,t):\Omega \times [0,\infty)\to\mathbb{R}$ is uniformly H\"older continuous with respect to $x$ and continuous with respect to $t$.
\end{itemize}
\begin{theorem}\label{1A1}
Assume $1<p<\frac{n+2s}{n-2s}$, $f$ satisfies
\begin{equation}\label{growth f}
    f(x,t)\le C_0(1+t^p)
\end{equation}
uniformly for all $x$ in $\Omega$, and
\begin{equation}\label{lim f}
    \lim _{t\to\infty}\frac{f(x,t)}{t^p} = K(x),
\end{equation}
    where $K(x)\in (0,\infty)$ is uniformly continuous and $K(\infty):=\lim _{|x|\to\infty }K(x) \in (0,\infty)$.
Then there exists a constant $C$, such that
\begin{equation}\label{bd1}
u(x)\le C,\quad\forall \, x\in \Omega
\end{equation}
holds for all nonnegative solutions $u$ of \eqref{main2}.
\end{theorem}

\begin{remark}
    The assumptions on $f$ are essential for the a priori estimate. When $f=0$ and $\Omega = B_1$, Abatangelo, Jarohs and Salda\~na \cite{Abatangelo2018Green} constructed nontrivial nonnegative solutions to \eqref{main2} that are unbounded, see also \cite{Li2023unique}.
\end{remark}

During the process of blow-up and rescaling, it can be shown that a sequence of the rescaled functions (still denoted by $u_i$) converges to a limiting function $u$ in the sense of $C^{2s+\beta}_{loc}$. In the case of a local operator, say $-\Delta$, we already have
\[   \lim _{i\to\infty}(-\Delta) u_i(x) = (-\Delta) u(x).\]
However, this is not the case for nonlocal operators, as shown in \cite[Theorem 1.1]{Du2023blowup} by Du-Jin-Xiong-Yang, in which they work on the whole space $\mathbb{R}^n$. In order to establish a priori estimate on general  unbounded domains with boundaries, we generalize  their convergence result as follows.
\begin{theorem}\label{conv}
    Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
    converges in $C^{2s+\beta}_{loc}(\Omega)$ to a function $u\in \mathcal{L}_{2s}$ , and $\{\mbox{$(-\triangle)^s$} u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
    \begin{equation}
        \lim _{i\to\infty}\mbox{$(-\triangle)^s$} u_i(x) = \mbox{$(-\triangle)^s$} u(x) -b.
    \end{equation}
\end{theorem}

 In Section 2, we establish local boundary H\"older regularity  for \eqref{main1} and prove Theorem \ref{1bdry C^s}. In Section 3, we derive  a priori estimates for \eqref{main2} and validate Theorem \ref{1A1}. Finally, the proof of Theorem \ref{conv} is included in the Appendix.

To prove Theorem~\ref{1bdry C^s}, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by Proposition \ref{otonserra}.” For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\alpha$-power decay near $\partial\Omega$:
\[
|h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\,\operatorname{dist}(x,\partial \Omega)^{\alpha},
\]
with $\alpha=\min\{s,1-s\}$. “Combining this decay estimate and the interior regularity result (Theorem \ref{Holder thm}),” one gets “the $C^\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.

To prove the main theorem, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by
\begin{proposition}[\cite{Rosoton2014Dirichlet}]\label{otonserra}
    Suppose $\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\in L^\infty(\Omega)$, $s\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:
    \begin{equation}\label{eq1}
        \begin{cases}
            \flap u = g,&\text{in } \Omega,\\
            u=0, &\text{in } \Omega ^c.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\R^n)$ and
\begin{equation}\label{hd}
    \norm{u}_{C^s(\R^n)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation}
where $C=C(n,s)$ is a positive constant.
\end{proposition}
For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\alpha$-power decay near $\partial\Omega$:
\[
|h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\,\operatorname{dist}(x,\partial \Omega)^{\alpha},
\]
with $\alpha=\min\{s,1-s\}$. Combining this decay estimate and the interior regularity result,
\begin{proposition}[Interior H\"older regularity]\label{Holder thm}
    Assume $0<s<1$, and $u\in C^{1,1}_{loc}(\R^n)\cap \mathcal{L}_{2s}$ is a nonnegative solution of
    \begin{equation}
        \flap u(x) = g(x),\quad x\in B_1
    \end{equation}
    where $g\in L^\infty(B_1)$. Then for any $0<\varepsilon <2s$,
    \begin{equation}
        \norm {u}_{C^{[2s-\varepsilon],\{2s-\varepsilon\}}(B_{1/2})}\le C(\norm{g}_{L^\infty(B_1)}+\norm{u}_{L^\infty(B_1)}).
    \end{equation}
\end{proposition}
one gets “the $C^\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.

Question:
Let \(\Omega\subset \mathbb{R}^n\) be a domain with locally \(C^{1,1}\) boundary, let \(0<s<1\), and fix a boundary point \(x_0\in \partial\Omega\). Write \(B_r:=B_r(x_0)\). Assume \(g\in L^\infty(\Omega\cap B_4)\), and let \(u\ge 0\) be a classical solution of
\[
\begin{cases}
(-\Delta)^s u=g,& \text{in }\Omega\cap B_4,\\
u=0,& \text{in }B_4\setminus \Omega,
\end{cases}
\]
where the fractional Laplacian is
\[
(-\Delta)^s v(x)=C_{n,s}\,\mathrm{P.V.}\int_{\mathbb{R}^n}\frac{v(x)-v(y)}{|x-y|^{n+2s}}\,dy.
\]
Which quantitative estimate holds for \(u\) up to the boundary in \(\Omega\cap B_{1/2}\)?

One has \(u\in C^{s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{s}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\bigr).
\]

Choices:
B.
One has \(u\in C^{s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{s}(\Omega\cap B_{1/2})}
\le C\,\|g\|_{L^\infty(\Omega\cap B_4)}.
\]
C.
One has \(u\in C^{\alpha_0}(\Omega\cap B_{1/2})\), where \(\alpha_0:=\min\{s,1-s\}\), and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{\alpha_0}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\bigr).
\]
D.
One has \(u\in C^{s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{s}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{\mathcal L_{2s}}\bigr).
\]
E.
One has \(u\in C^{2s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{2s}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\bigr).
\]


\begin{theorem}  \label{thm: factor hazard}
Let the hazard function, $h_0:(0,\infty)\to(0,\infty)$, admit the factorization:
\[
h_0(x)=C\,x^{p-1}\,g(x),\qquad C>0,\; p\ge 0,
\]
where $g:(0,\infty)\to(0,\infty)$ is non-decreasing. For $\tau>0$ define:
\[
r(x,\tau)\;=\;\tau\,\frac{h_0(\tau x)}{h_0(x)}.
\]
Then, for every $x>0$,
\[
0\le \tau\le 1
\quad\Longleftrightarrow\quad
0\le r(x,\tau)\le 1.
\]
\end{theorem}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0(\tau x)}{h_0(x)}, \qquad x>0.
\]
Then:
\[
0 \le r(x,\tau) \le 1 \quad \forall x>0
\quad \Longleftrightarrow \quad
0 \le \tau \le 1.
\]
\end{theorem}

\label{app: thms}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0

\begin{theorem} \label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. Then:
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]
\end{theorem}

As per \citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form:
\begin{align}
 \bar{F}_X(x) = P(X>x) = \bar{F}_0(x),\quad x>0, \label{eq: general marginal}
\end{align}
for some survival function $\bar{F}_0(x) \in [0, 1] \ \forall x> 0$, and for each $x>0$:
\begin{equation}
 P(Y>y\mid X>x) = \bar{F}_1\left(\beta(x)y\right), \quad x, y >0,\label{eq: general cond}
\end{equation}
for some survival function $\bar{F}_1(\beta(x)y) \in [0, 1] \ \forall x, y >0$ and a suitable acceleration function $\beta(x)$. As per \citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\{X > x\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:
\begin{align}
 P(X>x, Y>y) = \bar{F}_0(x) \bar{F}_1\left(\beta(x)y\right), \quad x, y>0. \label{eq: general joint}
\end{align}
Assuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:
\[
f_X(x) = -\frac{d}{dx}P(X>x) = -\frac{d}{dx}\bar{F}_0(x) = f_0(x),
\]
and since $\lim_{x\to 0^+}\bar{F}_0(x)=1$, where $\beta(0):=\lim_{x\to 0^+}\beta(x)$:
\[
P(Y>y) = \lim_{x\to 0^+}P(Y>y,X>x) = \bar{F}_1\bigl(\beta(0)y\bigr).
\]
Now, the density of $Y$ is:
\[
f_Y(y) = -\frac{d}{dy}P(Y>y)
= -\frac{d}{dy}\bar{F}_1(\beta(0)y)
= \beta(0)\,f_1(\beta(0)y),
\]
where $f_1(t)=-\bar{F}_1'(t)$ denotes the density associated with $\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\bar{F}_1$, but with its argument scaled by $\beta(0)$. For \eqref{eq: general joint} to be a valid survival function, it must have a non-negative
mixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\bar{F}_0$ and $\bar{F}_1$, and differentiating, we obtain:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) &=
f_0(x) f_1(\beta(x) y)\beta(x)
- \bar{F}_0(x) f_1'(\beta(x) y) \beta(x) \beta'(x) y
- \bar{F}_0(x) f_1(\beta(x) y) \beta'(x) \nonumber \\
&= f_0(x) f_1(\beta(x) y)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(\beta(x) y) + f_1'(\beta(x) y) \beta(x) y\right]. \nonumber
\end{align}
Denoting $t = \beta(x)y$, we have:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y)
&= f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(t) + f_1'(t)t\right] \nonumber \\
& = f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ t f_1(t) \right]' .\nonumber
\end{align}
Thus a necessary and sufficient condition for $ \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) \geq 0$ is:
\[
\beta'(x)\;\begin{cases}
\le\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'>0,\\[10pt]
\ge\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'<0, \\
\text{unconstrained}, & \text{if } [t f_1(t)]'=0.
\end{cases}
\]
Denoting the hazard function $h_0(x) = \frac{f_0(x)}{\bar{F}_0(x)}$ and $S(t) = \frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:
\begin{align}
\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\beta(x)
\;\leq\; \beta'(x)
\;\leq\;
\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\beta(x). \label{eq: B'(x) bounds}
\end{align}
A special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \frac{f_1(t)}{\left[tf_1(t) \right]'} = \frac{f_1(t)}{f_1(t) + tf_1'(t) } \geq1$ for $t \geq 0$, hence $\beta'(x)
\;\leq\;
 h_0(x)\beta(x)$.\\

It should be noted that multiple functional forms for $\beta(x)$ are possible. The question of selecting a form of $\beta(x)$ that provides the best model fit has already been investigated by \citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\beta(x)$ that arise from the extreme case in which $\beta'(x)$ attains its theoretical upper bound, that is, given $\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:
\begin{align}
 \beta'(x) &= c^*h_0(x) \beta(x), \nonumber \\
 \therefore \int \frac{1}{\beta(x)} d\beta(x) &= \int c^* h_0(x) dx, \nonumber\\
 \therefore \log\left( \beta(x) \right) &= -c^* \log\left( \bar{F}_0 (x)\right) + K. \nonumber \\
 \therefore \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(x)\right]^{c^*}}, \label{eq: beta(x) general}
 \end{align}
for $\gamma = e^K>0$. This suggests $\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\beta(0): = \lim_{x\to 0^+} \beta(x) = \gamma$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$. We also note that since $\bar{F}_0(x)$ approaches zero only as $x\rightarrow \infty$, the expression in \eqref{eq: beta(x) general} remains finite for all finite $x$.\\

The role of the acceleration function $\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.

\subsection{Moments}
Noting $X \sim Weibull(\alpha, \lambda)$ and $Y \sim Weibull(\gamma, \nu)$, we have:
\begin{align}
 E(X) &= \frac{1}{\alpha} \Gamma\left(1+ \frac{1}{\lambda}\right), \label{eq: Weibull ex} \\
 E(Y) &= \frac{1}{\gamma} \Gamma\left(1+ \frac{1}{\nu}\right), \label{eq: Weibull ey} \\
 {Var}(X) &= \frac{1}{\alpha^2}\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right),\label{eq: Weibull varx} \\
 {Var}(Y) &= \frac{1}{\gamma^2}\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right).\label{eq: Weibull vary}
\end{align}
Accordingly:
\begin{align}
\operatorname{Cov}(X, Y)
&= \int_0^\infty \int_0^\infty
 \big( P(Y>y , X> x) - P(X> x)P(Y> y) \big) \,dy\,dx \nonumber \\
&= \int_0^\infty \int_0^\infty
 \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr) \right) \,dy\,dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \int_0^\infty \left( e^{-(\beta(x)y )^\nu}- e^{-(\beta(0)y )^\nu} \right) dy \right) dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(x)} -\frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(0)} \right) dx \nonumber \\
&= \frac{1}{\gamma } \Gamma\!\left( 1+ \frac{1}{\nu}\right)
 \left[\int_0^\infty e^{-(\alpha x)^\lambda - \frac{1}{\nu}(\alpha \tau x)^\lambda} \, dx
 - \int_0^\infty e^{-(\alpha x)^\lambda} \, dx \right] \nonumber \\
&= \frac{1}{\alpha \gamma} \Gamma\!\left(1+ \frac{1}{\nu} \right)\Gamma\!\left(1+ \frac{1}{\lambda} \right)
 \left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right].
\label{eq: Weibull cov}
\end{align}
Now, using \eqref{eq: Weibull varx}, \eqref{eq: Weibull vary} and \eqref{eq: Weibull cov}, the correlation function is:
\begin{align}
 \rho(X,Y)
 &= \frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}} \nonumber\\
 &= \frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}}.
 \label{eq: Weibull rho}
\end{align}
Clearly, \eqref{eq: Weibull rho} is strictly decreasing in $\tau \in [0, 1]$. Hence:
\begin{align}
 \rho_{\text{min}}(\lambda, \nu):=\frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + 1} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}} \leq \rho(X, Y) \leq 0.
\end{align}
Now, $\inf_{\lambda, \nu > 0} \rho_{\text{min}}(\lambda, \nu) = \lim_{\nu \to \infty}\rho_{\text{min}}(\lambda = 1, \nu) = -\frac{\sqrt{6}}{\pi}$. Hence, this Weibull model with acceleration function $\beta(x) = \gamma e^{\frac{1}{\nu}(\alpha \tau x)^\lambda}$ will be able to accommodate correlations in the range $\rho(X,Y)\in\big[-\tfrac{\sqrt{6}}{\pi},\,0\big]$.

\appendix
\section*{Appendices}
\section{Joint Densities and Log-likelihoods} \label{app: joint lik}
\subsection{Exponential}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: exp joint survival}:
\[
f_{X,Y}(x,y)
= \alpha \gamma \,\bigl(\gamma \tau y e^{\alpha \tau x} - \tau + 1 \bigr)
 e^{\!\left(\alpha x(\tau - 1) - \gamma y e^{\alpha \tau x}\right)},
 \qquad x,y>0.
\]
where $\alpha, \gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\gamma,\tau)
&= n \log \alpha + n \log \gamma
+ \sum_{i=1}^n \log\!\Big( \gamma \tau y_i e^{\alpha \tau x_i} - \tau + 1 \Big) \nonumber \\
&\quad + \alpha (\tau - 1)\sum_{i=1}^n x_i
- \gamma \sum_{i=1}^n y_i e^{\alpha \tau x_i}. \nonumber
\end{align}
\subsection{Lomax}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Lomax joint survival}:
\begin{align*}
f_{X,Y}(x,y)
&= \nu \,(1+\alpha x)^{-3\lambda - 1}\,
 \bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{-3\nu - 4} \\[6pt]
&\quad \times \Biggl[
 \frac{\alpha \gamma^{2} \lambda \tau y (\nu+1)\,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{2\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+2}}
 {1+\alpha \tau x} \\[6pt]
&\qquad - \frac{\alpha \gamma \lambda \tau \,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}}
 {1+\alpha \tau x} \\[6pt]
&\qquad + \alpha \gamma \lambda \,\bigl((1+\alpha x)^{2}(1+\alpha \tau x)\bigr)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}
 \Biggr], \quad x,y >0,
\end{align*}
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \log \nu
 - (3\lambda+1)\sum_{i=1}^n \log(1+\alpha x_i) \nonumber \\
&\quad - (3\nu+4)\sum_{i=1}^n \log\!\bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr) \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Biggl\{
 \frac{\alpha \gamma^{2} \lambda \tau y_i (\nu+1)\,(1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{2\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+2} \nonumber \\
&\qquad\qquad
 - \frac{\alpha \gamma \lambda \tau (1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3} \nonumber \\
&\qquad\qquad
 + \alpha \gamma \lambda \,\bigl((1+\alpha x_i)^{2}(1+\alpha \tau x_i)\bigr)^{\lambda}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3}
 \Biggr\}. \nonumber
\end{align}
\subsection{Weibull}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Weibull joint survival}:
\[
f_{X, Y}(x,y) = \alpha^{\lambda} \gamma^{\nu} \lambda \nu x^{\lambda - 1} y^{\nu - 1}
\left( \tau^{\lambda} (\gamma y)^{\nu} e^{(\alpha \tau x)^\lambda}
- \tau^{\lambda} + 1 \right)
e^{(\alpha x)^\lambda(\tau^\lambda - 1)
- (\gamma y)^{\nu} e^{(\alpha\tau x)^{\lambda}}}, \quad x, y>0,
\]
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \lambda \log \alpha + n \nu \log \gamma
 + n \log \lambda + n \log \nu \nonumber\\
&\quad + (\lambda - 1) \sum_{i=1}^n \log x_i
 + (\nu - 1) \sum_{i=1}^n \log y_i \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Bigl(\tau^{\lambda} (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}
 - \tau^{\lambda} + 1 \Bigr) \nonumber \\
&\quad + (\tau^{\lambda} - 1)\sum_{i=1}^n (\alpha x_i)^\lambda
 - \sum_{i=1}^n (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}. \nonumber
\end{align}
\subsection{Log-logistic}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Log joint survival}:
\[
\begin{aligned}
f_{X,Y}(x,y)
&= \frac{\lambda \nu\,(\gamma y)^{\nu}}
{x\,y\,\bigl((\alpha x)^{\lambda}+1\bigr)^{2}
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)^{3}}
\\[4pt]
&\quad\times
\Biggl\{
(\alpha x)^{\lambda}\bigl((\alpha \tau x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
+ (\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
- 2\,(\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Biggr\},
\qquad x,y>0,
\end{aligned}
\]

\begin{proof}
Since $P(Y>y , X> x) = \bar{F}_0(x)\bar{F}_1\bigl(\beta(x)y\bigr)$ then $P(Y>y) = \lim_{x\to 0^+}P(Y>y , X> x) = \bar{F}_0(0)\bar{F}_1\bigl(\beta(0)y\bigr) = \bar{F}_1\bigl(\beta(0)y\bigr)$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$ and $\beta(0): = \lim_{x\to 0^+} \beta(x)$. We note from Hoeffding's covariance identity:
\begin{align}
    Cov(X, Y) & = \int_0^\infty \int_0^\infty\left( P(Y>y , X> x) - P(X> x)P(Y> y)   \right) \ dx dy \nonumber \\
    & = \int_0^\infty \int_0^\infty  \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr)  \right) \ dx dy.\nonumber
\end{align}
Now since $\bar{F}_1'(\cdot)\le 0$ (as $\bar{F}_1$ is a survival function), if $\beta'(x) \geq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \leq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \leq 0$ for $x, y \geq 0$. Conversely, if $\beta'(x) \leq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \geq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \geq 0$ for $x, y \geq 0$. Finally, if $\beta(x)$ is constant, say $\beta(x)\equiv c$, then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)\,P(Y>y)$, so $X$ and $Y$ are independent and ${Cov}(X,Y)=0$.
\end{proof}
Furthermore, as done in \citet{arnold2020bivariate}, we introduce a dependence parameter $\tau$ which controls the strength of the dependence between $X$ and $Y$. Extending from \eqref{eq: beta(x) general}, we define the acceleration function as such:
\begin{align}
    \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(\tau x)\right]^{c^*}}. \label{eq: beta(x) general 2}
\end{align}
Given the bounds established in \eqref{eq: B'(x) bounds}, and upon selecting a non-decreasing specification for $\beta(x)$ as defined in \eqref{eq: beta(x) general 2}, and again assuming $\inf_{t: [t f_1(t)]'>0} S(t) = c^* > 0$, we obtain bounds for $\tau$ as such:
\begin{align}
    0 &\leq \beta'(x) \leq c^* h_0(x)\, \beta(x), \nonumber\\
    \therefore
    0 &\leq
    \frac{c^* \tau \gamma f_0(\tau x)}{\big[\bar{F}_0(\tau x)\big]^{c^* + 1}}
    \leq
    c^* \gamma \,
    \frac{f_0(x)}{\big[\bar{F}_0(x)\big]^{c^*}}
    \frac{1}{\big[\bar{F}_0(\tau x)\big]^{c^*}}, \nonumber\\
    \therefore
    0 &\leq \tau \leq \frac{h_0(x)}{h_0(\tau x)}. \nonumber\end{align}
We re-write this inequality as such:
\begin{align}
    0 \le r(x, \tau) \le 1,
    \label{eq: tau bounds}
\end{align}
where $r(x, \tau) = \tau \frac{h_0(\tau x)}{h_0(x)}$, where the endpoint $\tau=0$ is included by continuity, with $r(x,0):=\lim_{\tau\to 0^+}r(x,\tau)=0$. Theorem \ref{thm: ABC} and Theorems \ref{thm: ifr hazard} and \ref{thm: factor hazard} in Appendix \ref{app: thms}, serve as a basis for establishing bounds on the dependence parameter $\tau$ that satisfy \eqref{eq: tau bounds}.

\begin{theorem}
\label{thm: ABC}
Let $h_0:(0,\infty)\to(0,\infty)$ be a hazard function. For $x>0$ and $\tau>0$ define:
\[
r(x,\tau):=\tau\,\frac{h_0(\tau x)}{h_0(x)},\qquad
\phi(x):=x\,h_0(x).
\]
Let
\[
A:\ \ 0\le r(x,\tau)\le 1\ \ \text{for all }x>0,\qquad
B:\ \ \tau\in(0,1],\qquad
C:\ \ \phi \text{ is non-decreasing on }(0,\infty).
\]
Then:
\begin{enumerate}
\item[(1)] $(A \wedge B)\ \Rightarrow\ C$.
\item[(2)] $(C \wedge B)\ \Rightarrow\ A$.
\item[(3)] $(C \wedge A)\ \Rightarrow\ B$, \emph{unless} $\phi$ is identically constant on $(0,\infty)$, in which case $A$ holds for all $\tau>0$ and $B$ cannot be inferred.
\end{enumerate}

\end{theorem}

\begin{proof}
\textbf{(1) $(A \wedge B)\Rightarrow C$.}
Fix $0<y<x$ and set $\tau:=y/x\in(0,1]$ so that $B$ holds. By $A$,
\[
1 \ \ge\ r(x,\tau)=\tau\,\frac{h_0(\tau x)}{h_0(x)}=\frac{y\,h_0(y)}{x\,h_0(x)}.
\]
Hence $y\,h_0(y)\le x\,h_0(x)$ for all $0<y<x$, i.e.\ $\phi$ is non-decreasing. Thus $C$ holds.

\medskip
\textbf{(2) $(C \wedge B)\Rightarrow A$.}
Assume $C$ and $\tau\in(0,1]$. Then $\tau x\le x$ and, by the non-decreasing nature of $\phi$,
\[
\phi(\tau x)\le \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x)\le x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau)\le 1.
\]
Since $h_0>0$, we also have $r(x,\tau)\ge 0$. Hence $A$ holds.

\medskip
\textbf{(3) $(C \wedge A)\Rightarrow B$, unless $\phi$ is constant.}
Assume $C$ and $A$ hold. Because $h_0(x)>0$, the lower bound $r(x,\tau)\ge 0$ implies $\tau \ge 0$. Now suppose, toward a contradiction, that $\tau>1$, then $\tau x > x$ and, by assuming $\phi$ is now strictly increasing:
\[
\phi(\tau x) > \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x) > x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau) > 1.
\]
contradicting the upper bound. Hence $\tau \le 1$. If instead $\phi$ is identically constant,
say $\phi(x)\equiv K>0$, then $h_0(x)=K/x$ and
\[
r(x,\tau)=\tau\,\frac{K/(\tau x)}{K/x}=1 \quad (\forall x>0,\ \forall \tau>0),
\]
so $A$ holds for all $\tau>0$ and no restriction $B$ follows.
\end{proof}
\begin{remark} When convenient, extend $B$ to $\tau\in[0,1]$ by defining $r(x,0):=\lim_{\tau\to 0^+} r(x,\tau)=0$ (whenever the limit exists). All implications above then carry over with this convention.
\end{remark}
\begin{remark}
If $h_0$ is non-decreasing, then $\phi(x) =  x h_0(x)$ is non-decreasing (product of increasing, positive functions), so Theorem~\ref{thm: ABC} recovers Theorem \ref{thm: ifr hazard}.
If $h_0(x)=C\,x^{p-1}g(x)$ with $C>0$, $p\ge 0$, and $g(x)$ non-decreasing, then $\phi(x) = C x^pg(x)$ is non-decreasing, so Theorem \ref{thm: ABC} recovers Theorem \ref{thm: factor hazard}.
\end{remark}

The following sections present the range of models employed in this study. In each case, the survival function forms are specified according to a single particular rate-shape distributional family such that $X \sim Family (\alpha, \lambda)$ and $Y \sim Family (\beta(0), \nu)$ where $\alpha$ and $\beta(0)$ denote the rate parameters and $\lambda$ and $\nu$ denote the shape parameters (shape parameters are not applicable for exponential and half-Cauchy models, however). Closed-form expressions of the models' moments are derived, and additionally, theoretical correlation bounds are obtained (with the exception of the half-Cauchy and gamma models).

Using $P(Y>y,X>x)=\bar F_0(x)\,\bar F_1(\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\to 0^+$:
\[
P(Y>y)=\lim_{x\to 0^+}P(Y>y,X>x)=\bar F_1(\beta(0)y),\qquad \beta(0):=\lim_{x\to 0^+}\beta(x),
\]
(since $\lim_{x\to 0^+}\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:
\[
\operatorname{Cov}(X,Y)=\int_0^\infty\!\int_0^\infty \Big(P(Y>y,X>x)-P(X>x)P(Y>y)\Big)\,dx\,dy
=\int_0^\infty\!\int_0^\infty \bar F_0(x)\Big(\bar F_1(\beta(x)y)-\bar F_1(\beta(0)y)\Big)\,dx\,dy.
\]
Since $\bar F_1'(\cdot)\le 0$, if $\beta'(x)\ge 0$ then $\bar F_1(\beta(x)y)\le \bar F_1(\beta(0)y)$, making the integrand non-positive for $x,y\ge 0$, hence $\operatorname{Cov}(X,Y)\le 0$; conversely, if $\beta'(x)\le 0$ then $\bar F_1(\beta(x)y)\ge \bar F_1(\beta(0)y)$, so $\operatorname{Cov}(X,Y)\ge 0$. Finally, if $\beta(x)\equiv c$ is constant then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\operatorname{Cov}(X,Y)=0$.

\label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. To prove the main theorem, it suffices to show that
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]

Question: Let \(X\) and \(Y\) be positive random variables in an accelerated conditional model with marginal survival function \(P(X>x)=\bar F_0(x)\) and conditional survival function \(P(Y>y\mid X>x)=\bar F_1(\beta(x)y)\) for \(x,y>0\), where \(\bar F_0\) and \(\bar F_1\) are survival functions and \(\beta\) is the acceleration function. Which sign relationship between the monotonicity of \(\beta\) and the covariance \(\operatorname{Cov}(X,Y)\) holds?

If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\le 0\); if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\).

Choices:
B. If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\); if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\le 0\).
C. If \(\beta(x)\) is constant for all relevant \(x\), then \(\operatorname{Cov}(X,Y)=0\).
D. If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)< 0\); if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)> 0\).
E. If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(X\) and \(Y\) are independent; if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\).

monotonicity of survival function flips covariance sign

dropped the full monotonicity-to-sign conclusion, retaining only the constant-\(\beta\) independence case

non-strict inequalities from pointwise sign of Hoeffding integrand

constant-\(\beta\) is required for factorization, not merely monotone increasing \(\beta\)


\label{main-2}  Let $R$ be a finite family of slopes containing $0,1,\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.
\begin{itemize}
    \item[(i)] (Three slopes) If $k=0$, then
\begin{equation}\label{2ab} 2 - \frac{c_2}{D(R;s)} \leq \SD\left(R; s\right) \leq 2 - \frac{c_1}{D(R;s)}
\end{equation}
    for some absolute constants $c_2 > c_1 > 0$.
    \item[(ii)]  (Many slopes) In general, one has
    \begin{equation}\label{sam} 2 - \frac{C_k \log(2+D(R;s))}{D(R;s)} \leq \SD(R; s) \leq 2 - \frac{c_k}{D(R;s)^{k+1}}
    \end{equation}
    for some absolute constants $c_k, C_k > 0$.
\end{itemize}

\begin{equation}\label{dxy}
d[X;Y], d[Y;Y] \ll \log K.
\end{equation}

For $a \in \Q$, define $g(a) \coloneqq \H[Y-aY'] - \H[Y]$, where $Y'$ is an independent copy of $Y$. From \eqref{hx}, \eqref{ruzsa}, \eqref{ruzsa-diff} we have $g(r) \ll \log K$ for $r \in R \backslash \{\infty\}$.  By Proposition \ref{dilate}(iv), we conclude that $g(s) \ll D^{k+1} \log K$, thus $d[Y; sY] \ll D^{k+1} \log K$.  Combining with \eqref{dxy} and \eqref{ruzsa}, \eqref{ruzsa-diff}, we obtain \eqref{dxsy} as required.

\section{A continuous limit}\label{cts-sec}

In this section we establish \Cref{cts-limit}.  Let $\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\infty$ be as in that theorem.  We allow implied constants in the asymptotic notation to depend on $\alpha, f$. We will establish the lower bound
\begin{equation}\label{sd-targ}
 \SD\left(\{0,1,\infty\},\frac{a}{b}\right) \geq 2 - \frac{h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))+o(1)}{\log b}.
 \end{equation}

\subsection{The arithmetic Kakeya conjecture}

Define a \emph{slope} to be an element $r$ of the projective rational\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic.  But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\Q \cup \{\infty\}$.  We then define the projection operators $\pi_r \colon \Q \times \Q \to \Q$ by setting
$$ \pi_r(x,y) \coloneqq x + ry$$
for $r \neq \{\infty\}$, and
$$ \pi_\infty(x,y) \coloneqq y.$$
Given a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \emph{sum-difference constant} $\SD(R;s)$ to be the least exponent such that the bound
\begin{equation}\label{sdef}
\H[ \pi_s(X,Y) ] \leq \SD(R;s) \max_{r \in R} \H[\pi_r(X,Y)]
\end{equation}
holds for $\Q$-valued random variables\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where
$$ \H[X] \coloneqq \sum_{x} \h(\P(X=x))$$
is the Shannon entropy of a random variable $X$, with $\h(t) \coloneqq t \log \frac{1}{t}$ using the convention $\h(0) \coloneqq 0$.  Thus for instance
$$\H[ X-Y ] \leq \SD(\{0,1,\infty\};-1) \max\left( \H[X], \H[X+Y], \H[Y] \right)$$
whenever $X,Y$ are (possibly dependent) $\Q$-valued random variables.

The quantity $\SD(R;s)$ can equivalently be defined as the least exponent such that the bound
\begin{equation}\label{pise}
|\pi_{s}(E)| \leq (\max_{r \in R} |\pi_s(E)|)^{\SD(R;s)}
\end{equation}
for all finite non-empty $E \subset \Q$, where $|A|$ denotes the cardinality of a finite set $A$; see \cite{green}.  However, it will be convenient in this paper to work with the entropy formulation, in order to take advantage of the ``entropic Pl\"unnecke--Ruzsa calculus'' that are founded on the Shannon entropy inequalities.

It is easy to see that one has the projective invariance
$$ \SD(\phi(R); \phi(s)) = \SD(R; s)$$
for any projective transformation $\phi \colon \Q \cup \{\infty\} \to \Q \cup \{\infty\}$, that is to say a map of the form $\phi(r) \coloneqq \frac{ar+b}{cr+d}$ for some $a,b,c,d \in \Q$ with $ad-bc \neq 0$, with the usual conventions when $r$ is infinite or $cr+d$ vanishes.  For instance, by using a dilation transformation, we have
$$ \SD(\{0,1,\infty\}; s) = \SD(\{0,-1/s,\infty\}; -1)$$
for any slope $s$ other than $0,1,\infty$.
In the literature it is conventional to use this $3$-transitive projective symmetry to normalize $s = -1$ and $0, \infty \in R$ (assuming that $|R|\geq 2$ of course), though in this paper it will be more convenient to adopt the normalization $0,1,\infty \in R$ (assuming $|R| \geq 3$).

It is easy to see that $\SD(R;s)=\infty$ when $|R|<2$.
From the entropy inequality $\H[X-Y] \leq \H[X,Y] \leq \H[X]+\H[Y]$ we see that $\SD(R;-1) \leq 2$ when $0,\infty \in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \{0,\infty\}$.  By projective invariance, this implies that $\SD(R;s)=2$ whenever $|R|=2$.  As these quantities are clearly non-decreasing in $R$, we then have the trivial bound
\begin{equation}\label{sdf-upper}
\SD(R;s) \leq 2
\end{equation}
for $|R| \geq 2$.

Improvements upon \eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions.  Indeed, in \cite{bourgain} it was observed that a bound of the form $\SD(R;-1) \leq \alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\'edi's theorem by Leng, Sawhney, and Sah \cite{lss}, that one also obtains this bound for the Hausdorff dimension as well.  For the corresponding results for packing dimension, see \cite{cowen}.} at most $\frac{d-1}{\alpha}+1$.  In particular, if one can establish the \emph{arithmetic Kakeya conjecture}
\begin{equation}\label{infr}
\inf_R \SD(R;-1) = 1
\end{equation}
then this would imply that Kakeya and Nikodym sets in $\R^d$ have full Minkowski and Hausdorff dimension for all $d$.  This is currently only known for $d \leq 3$ \cite{wang}. We refer the reader to \cite{green}, \cite{cowen}, \cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.

Nontrivial progress towards the arithmetic Kakeya conjecture was first obtained in  \cite{bourgain}, who in our notation showed that
$$ \SD(\{0,1,\infty\};-1)  \leq 2 - \frac{1}{13} = 1.923\dots,$$
and used this to obtain new bounds on the Kakeya conjecture in high dimensions.
Further improvements were then obtained in \cite{katz-tao}, \cite{katz-tao-new}. For instance, it is known that
$$ 1.77898 \leq \SD(\{0,1,\infty\};-1) \leq 2 - \frac{1}{6} = 1.833\dots$$
and
$$ 1.668 \leq \SD(\{0,1,2,\infty\};-1) \leq 2 - \frac{1}{4} = 1.75,$$
with the upper bounds established in \cite{katz-tao}, and the lower bounds in \cite{lemm}, \cite{gdm} respectively.
At present, the best upper bound known towards \eqref{infr} is
$$\inf_R \SD(R;-1) \leq 1.67513\dots;$$
see \cite{katz-tao-new}.

\subsection{Asymptotic behavior}

Informally, the arithmetic Kakeya conjecture asserts that in the asymptotic regime where the number of slopes $R$ is large, the constants $\SD(R;s)$ converge to $1$.  Here we consider a complementary regime, in which the number of slopes $R$ is fixed, but we instead let the elements of $R$ (or $s$) vary.  Our main results assert, roughly speaking, that the behavior of these constants is determined by the \emph{rational complexity} of $s$ relative to $R$, with the constants approaching $2$ as it becomes harder to express $s$ in terms of a rational expression of the $R$.  We give a (slightly artificial) definition of this quantity, restricting attention to the normalized setting $\{0,1,\infty\} \subset R$ for simplicity.

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

Informally, if the complexity of $s$ with respect to $R$ is equal to $D$, then $s$ can be expressed in terms of the slopes in $R$ by a rational expression whose length (when expressed as string of characters) is comparable to $D$.  The rational complexity is reminiscent of the \emph{arithmetic circuit complexity} of $s$ in terms of $R$, but with the key difference that the circuit must take the specific rational form \eqref{rat}.

\begin{example}  $D(\{0,1,\infty\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$.  In particular, if $a,b$ are coprime then
$$D \left( \{0,1,\infty\};\frac{a}{b} \right) \asymp \log(2+|a|+|b|).$$
\end{example}

Our main result, proven in Section \ref{main-sec}, is then as follows.

Define a \emph{slope} to be an element $r$ of the projective rational\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\Q \cup \{\infty\}$. We then define the projection operators $\pi_r \colon \Q \times \Q \to \Q$ by setting
$$ \pi_r(x,y) \coloneqq x + ry$$
for $r \neq \{\infty\}$, and
$$ \pi_\infty(x,y) \coloneqq y.$$
Given a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \emph{sum-difference constant} $\SD(R;s)$ to be the least exponent such that the bound
\begin{equation}\label{sdef}
\H[ \pi_s(X,Y) ] \leq \SD(R;s) \max_{r \in R} \H[\pi_r(X,Y)]
\end{equation}
holds for $\Q$-valued random variables\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where
$$ \H[X] \coloneqq \sum_{x} \h(\P(X=x))$$
is the Shannon entropy of a random variable $X$, with $\h(t) \coloneqq t \log \frac{1}{t}$ using the convention $\h(0) \coloneqq 0$. Thus for instance
$$\H[ X-Y ] \leq \SD(\{0,1,\infty\};-1) \max\left( \H[X], \H[X+Y], \H[Y] \right)$$
whenever $X,Y$ are (possibly dependent) $\Q$-valued random variables.

It is easy to see that $\SD(R;s)=\infty$ when $|R|<2$.
From the entropy inequality $\H[X-Y] \leq \H[X,Y] \leq \H[X]+\H[Y]$ we see that $\SD(R;-1) \leq 2$ when $0,\infty \in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \{0,\infty\}$. By projective invariance, this implies that $\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound
\begin{equation}\label{sdf-upper}
\SD(R;s) \leq 2
\end{equation}
for $|R| \geq 2$.

Improvements upon \eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \cite{bourgain} it was observed that a bound of the form $\SD(R;-1) \leq \alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\'edi's theorem by Leng, Sawhney, and Sah \cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \cite{cowen}.} at most $\frac{d-1}{\alpha}+1$. In particular, if one can establish the \emph{arithmetic Kakeya conjecture}
\begin{equation}\label{infr}
\inf_R \SD(R;-1) = 1
\end{equation}
then this would imply that Kakeya and Nikodym sets in $\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \leq 3$ \cite{wang}. We refer the reader to \cite{green}, \cite{cowen}, \cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

\begin{example} $D(\{0,1,\infty\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then
$$D \left( \{0,1,\infty\};\frac{a}{b} \right) \asymp \log(2+|a|+|b|).$$
\end{example}

Our main result, proven in Section \ref{main-sec}, is then as follows.

\label{main-sec}

We now prove Theorem \ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \geq 1$ and $a \neq 0,b$ is coprime to $b$, then $D(R;s) \asymp \log(2+|a|+|b|)$.  Let $N$ be the

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

Our main result, proven in Section \ref{main-sec}, is then as follows.

\begin{figure}
 \centering
\centerline{\includegraphics[width=\linewidth]{best.png}}
 \caption{Lower bounds on $\SD(\{0,1,\infty\}; s)$ obtained by \texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}
 \label{fig:kak}
\end{figure}

\begin{theorem}[Continuous limit]\label{cts-limit} Let $\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \to \infty$ and $a/b \to \alpha$. Then
\begin{equation}\label{up}
 \SD\left(\{0,1,\infty\};\frac{a}{b}\right) \geq 2 - \frac{c_\alpha+o(1)}{\log b}
 \end{equation}
 as $b\to \infty$ and $a/b \to \alpha$, where $0 < c_\alpha < \infty$ is the infimum of all quantities
\begin{equation}\label{hf}
 h_2(f_{(\alpha)}) - \max(h(f_0), h(f_\infty), h(f_\alpha))
 \end{equation}
where $f \colon \R^2 \to \R^+$ ranges over smooth compactly supported functions with total mass $\int_{\R^2} f(x,y)\ dx dy$ equal to $1$, $f_{(\alpha)} \colon \R \times [0,1) \to \R^+$ is a ``folded'' version
\begin{equation}\label{falpha-def}
 f_{(\alpha)}(x,y) \coloneqq \sum_{j \in \Z} f(x-\alpha j, y+j)
 \end{equation}
of $f$, $f_0, f_\infty, f_1 \colon \R \to \R^+$ are the projections
\begin{align}
f_0(x) &\coloneqq \int_\R f(x,y)\ dy\label{f0-def}\\
f_\infty(y) &\coloneqq \int_\R f(x,y)\ dx\label{finfty-def}\\
f_1(z) &\coloneqq \int_\R f(x,z- x)\ dx,\label{f1-def}
\end{align}
the two-dimensional differential entropy $h_2(f_{(\alpha)})$ of $f_{(\alpha)}$ is defined as
$$h_2(f_{(\alpha)}) \coloneqq \int_0^1 \int_\R \h(f_{(\alpha)}(x,y))\ dx dy,$$
and the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\infty$ is defined as
$$ h(f_r) \coloneqq \int_\R \h(f_r(x))\ dx.$$
\end{theorem}

\begin{proposition}\label{dilate} Let $X$ be an $\Q$-valued random variable, and for any $a \in \Q$, let $g(a)$ denote the quantity $g(a) \coloneqq \H[X-aX'] - \H[X]$, where $X'$ is an independent copy of $X$.
\begin{itemize}
 \item[(i)] $g(0) = 0$ and $g(1)=d[X;X]$.
 \item[(ii)] For any $a \in \Q$, we have $g(-a) \leq 3g(a)$, and if $a$ is non-zero, $g(a^{-1}) = g(a)$.
 \item[(iii)] For any $k \geq 1$ and $a_1,\dots,a_k \in \Q$, we have $g(a_1 \dots a_k) \leq g(a_1)+ \dots + g(a_k)$ and $g(a_1 + \dots + a_k) \leq g(a_1)+\dots+g(a_k)+(k-1) g(1)$.
 \item[(iv)] If $a$ is a non-zero integer, then $g(a) \leq (4 + 10 \lfloor \log_2 |a| \rfloor) g(1)$.
 \item[(v)] If $R$ is a finite set of slopes containing $0,1,\infty$ of cardinality $k+3$, then
 $$ g(a) \ll D(R;a)^{k+1} \max_{r \in R \backslash \{\infty\}} g(r).$$
\end{itemize}
\end{proposition}

In this section we establish \Cref{cts-limit}. Let $\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\infty$ be as in that theorem. We allow implied constants in the asymptotic notation to depend on $\alpha, f$. We will establish the lower bound
\begin{equation}\label{sd-targ}
 \SD\left(\{0,1,\infty\},\frac{a}{b}\right) \geq 2 - \frac{h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))+o(1)}{\log b}.
 \end{equation}
Comparing this with Theorem \ref{main-2}, we conclude that the expression $h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))$ is bounded from below, hence $c_\alpha > 0$; by taking an arbitrary test function for $f$ we also see that $c_\alpha < \infty$. Taking $h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))$ arbitrarily close to $c_\alpha$, we obtain the desired claim \eqref{up}.

It remains to establish \eqref{sd-targ}.
By the Poisson summation formula, the rapid decrease of the Fourier transform of the smooth compactly supported $f$, and the mass one hypothesis, we have
\begin{equation}\label{bib}
 \frac{1}{b^2} \sum_{n,m \in \Z} f\left(\frac{n}{b}, \frac{m}{b}\right) = 1 + \eps
 \end{equation}
for some $\eps = O(1/b)$; in fact one can get much better decay than this, but for our purposes any decay faster than $1/\log b$ will suffice. We then take $X,Y$ to be supported on the grid $\Z^2$ with probability distribution
$$ \P((X,Y) = (n,m)) \coloneqq \frac{1}{(1+\eps) b^2} \left(\frac{n}{b}, \frac{m}{b}\right)$$
thus $(X,Y)$ takes values in a ball of radius $O(b)$ and
\begin{equation}\label{pnx}
\P((X,Y) = (n,m)) = \frac{1}{b^2} f\left(\frac{n}{b}, \frac{m}{b}\right) + O\left( \frac{1}{b^3} \right).
\end{equation}
By Bezout's theorem, any integer can be uniquely written in the form $bn+am$ for some $0 \leq m < b$, and any other representation of the form $bn'+am'$ takes the form $b(n-aj) + a(m+bj)$. Thus
$$ \P(bX+aY = bn+am) = \sum_{j \in\Z} \P( (X,Y) = (n-aj, m+bj) ).$$
There are only $O(1)$ values of $j$ for which this sum is non-zero, so from \eqref{pnx} one has
$$ \P(bX+aY = bn+m) = \frac{1}{b^2} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) + O\left( \frac{1}{b^3} \right).$$
where $f_{(a,b)}: \R \times [0,1) \to \R^+$ is the function
$$f_{(a/b)}(x,y) \coloneqq \sum_{j \in \Z} f\left(x-\frac{a}{b} j, y+j\right).$$
Applying the entropy function $\h$, we conclude that
$$ \h(\P(bX+aY = bn+am)) = \frac{2 \log b}{b^2} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) +
\frac{1}{b^2} \h\left(f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right)\right) + O\left( \frac{\log b}{b^3} \right).$$
From \eqref{bib} one has
$$ \frac{1}{b^2} \sum_{n \in\Z; 0 \leq m < b} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) = 1 + \eps = 1 + O\left(\frac{1}{b} \right)$$
and from (uniform) Riemann integrability of the $f_{(a/b)}$ one has
$$ \frac{1}{b^2} \sum_{n \in\Z; 0 \leq m < b} \h\left(f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right)\right) = h_2(f_{(a/b)}) + o(1).$$
Finally, from dominated convergence one has
$$ h_2(f_{(a/b)}) = h_2(f_{(\alpha)})+o(1)$$
so we conclude that
$$ \H\left[\pi_{a/b}(X,Y)\right] = \H[bX+aY] = 2 \log b + h_2(f_{(\alpha)}) + o(1).$$
In a similar vein, from another application of Poisson summation and \eqref{f0-def} we see that
$$ \P(X=n) = \frac{1}{b} f_0\left(\frac{n}{b}\right) + O\left(\frac{1}{b^2}\right)$$
for any integer $n$, hence
$$ \h(\P(X=n)) = \frac{\log b}{b} f_0\left(\frac{n}{b}\right) + \frac{1}{b} \h\left(f_0 \left(\frac{n}{b}\right)\right) + O\left(\frac{\log b}{b^2}\right).$$
Since $X = O(b)$, one can sum using Riemann integrability and \eqref{bib} to conclude that
$$ \H[\pi_0(X,Y)] = \H[X]= \log b + h(f_0) + o(1).$$
Similar arguments give
$$ \H[\pi_\infty(X,Y)] = \H[Y] = \log b + h(f_\infty) + o(1)$$
and
$$ \H[\pi_1(X,Y)] = \H[X+Y] = \log b + h(f_1) + o(1).$$
Comparing this with \eqref{sdef}, we obtain \eqref{sd-targ}.

\begin{figure}
    \centering
\centerline{\includegraphics[width=\linewidth]{best.png}}
    \caption{Lower bounds on $\SD(\{0,1,\infty\}; s)$ obtained by \texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \log(|a|+b|)$.  The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}
    \label{fig:kak}
\end{figure}

The logarithmic convergence in \eqref{2ab} was suggested to us by experiments in \cite{gdm} using \texttt{AlphaEvolve} to obtain lower bounds for $\SD(\{0,1,\infty\}; s)$ for various slopes $s$.  This data was of low accuracy, as \texttt{AlphaEvolve} could only provide lower bounds and not upper bounds for these quantities; nevertheless, a logarithmic decay was numerically evident (see Figure \ref{fig:kak}), and furthermore the approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$ obtained by this tool (see \cite[Figure 18]{gdm}) suggested an approach to make the lower bound in \eqref{2ab} rigorous.  Once this was accomplished, the author was also able to obtain the matching upper bound in \eqref{2ab} using the entropic Pl\"unnecke--Ruzsa calculus.   A modification of these arguments, with some inefficiencies, then gives \eqref{sam}. We tentatively conjecture that the bounds in \eqref{sam} can be improved to be of the form \eqref{2ab} for all $k$, not just $k=0$ (possibly after some slight adjustments to the definition of rational complexity).

In the limit as $a/b$ converges to some real number $\alpha$ and $b \to \infty$, we can obtain a more precise lower bound (also suggested by the aforementioned \texttt{AlphaEvolve} numerics) as follows.

\begin{theorem}[Continuous limit]\label{cts-limit}  Let $\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \to \infty$ and $a/b \to \alpha$.  Then
\begin{equation}\label{up}
 \SD\left(\{0,1,\infty\};\frac{a}{b}\right) \geq 2 - \frac{c_\alpha+o(1)}{\log b}
 \end{equation}
 as $b\to \infty$ and $a/b \to \alpha$, where $0 < c_\alpha < \infty$ is the infimum of all quantities
\begin{equation}\label{hf}
 h_2(f_{(\alpha)}) - \max(h(f_0), h(f_\infty), h(f_\alpha))
 \end{equation}
where $f \colon \R^2 \to \R^+$ ranges over smooth compactly supported functions with total mass $\int_{\R^2} f(x,y)\ dx dy$ equal to $1$, $f_{(\alpha)} \colon \R \times [0,1) \to \R^+$ is a ``folded'' version
\begin{equation}\label{falpha-def}
 f_{(\alpha)}(x,y) \coloneqq \sum_{j \in \Z} f(x-\alpha j, y+j)
 \end{equation}
of $f$, $f_0, f_\infty, f_1 \colon \R \to \R^+$ are the projections
\begin{align}
f_0(x) &\coloneqq \int_\R f(x,y)\ dy\label{f0-def}\\
f_\infty(y) &\coloneqq \int_\R f(x,y)\ dx\label{finfty-def}\\
f_1(z) &\coloneqq \int_\R f(x,z- x)\ dx,\label{f1-def}
\end{align}
the two-dimensional differential entropy $h_2(f_{(\alpha)})$  of $f_{(\alpha)}$ is defined as
$$h_2(f_{(\alpha)}) \coloneqq \int_0^1 \int_\R \h(f_{(\alpha)}(x,y))\ dx dy,$$
and the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\infty$ is defined as
$$ h(f_r) \coloneqq \int_\R \h(f_r(x))\ dx.$$
\end{theorem}

We establish this result in \Cref{cts-sec}.
We tentatively conjecture that the upper bound in \eqref{up} is in fact an asymptotic equality, so that the asymptotic behavior of
$ \SD\left(\{0,1,\infty\}, \frac{a}{b}\right)$ is controlled not only by the rational complexity (as represented by the $\log b$ denominator), but also by the variational quantity $c_\alpha$ appearing in the numerator.  It is not clear whether this quantity $c_\alpha$ can be computed exactly; numerically, two-dimensional gaussians are reasonably good candidates for $f$, but in practice they do not extremize the functional \eqref{hf} precisely.

\subsection{Notation}

We use the asymptotic notation $X = O(Y)$, $X \ll Y$, or $Y \gg X$ to denote the assertion that $|X| \leq CY$ for some absolute constant $C$; if we need this implied constant $C$ to depend on some fixed quantities (such as the number $k$ of slopes), we will indicate this in the text.

When there is no possibility of ambiguity, we omit parentheses from pairs $(X,Y)$ of random variables, for instance abbreviating $\H[(X,Y)]$ as $\H[X,Y]$.  Given a random variable $Y$ taking values in some set $S$ and some function $f \colon S \to \R$, we define the expectation
$$ \E_{Y=y} f(y) \coloneqq \sum_y \P(Y=y) f(y)$$
where $y$ ranges over the essential range of $Y$.  While this expression could also be abbreviated as $\E f(Y)$, it will be notationally useful to distinguish between the random variable $Y$ and the possible values $y$ that this variable could take.  For instance, with this notation, the \emph{conditional entropy} $\H[X|Y]$ of one random variable $X$ with respect to another $Y$ can now be defined by the formula
$$ \H[X|Y] \coloneqq \E_{Y=y} \H[ X | Y = y ]$$
where $(X|Y=y)$ is $X$ conditioned to the event $Y=y$ (again, we omit parentheses when there is no possibility of ambiguity).  The chain rule asserts that $\H[X|Y]$ can also be expressed by the formula
$$ \H[X|Y] = \H[X,Y] - \H[Y].$$
The \emph{mutual information} $\I(X:Y)$ between two random variables is given by the formula
$$ \I(X:Y) = \H[X] - \H[X|Y] = \H[Y] - \H(Y|X) = \H[X] + \H[Y] - \H[X,Y].$$
As is well known, $\I(X:Y)$ is non-negative, and vanishes precisely when $X,Y$ are independent.  Equivalently, one has the subadditivity property
$$ \H[X,Y] \leq \H[X] + \H[Y]$$
with equality precisely when $X,Y$ are independent.

We also define the conditional mutual information
$$ \I[X:Y|Z] \coloneqq \E_{Z=z} \I[(X|Z=z):(Y|Z=z)].$$
Clearly, $\I[X:Y|Z]$ is non-negative, and vanishes precisely when $X,Y$ are independent conditionally on $Z$.
From the chain rule we have
\begin{equation}\label{i-split}
\begin{split}
\I[X:Y|Z] &= \H[X|Z] - \H[X|Y,Z] \\
&= \H[Y|Z] - \H[Y|X,Z] \\
&= \H[X|Z] + \H[Y|Z] - \H[X,Y|Z].
\end{split}
\end{equation}

\subsection{Acknowledgments}

The author was supported by the James and Carol Collins Chair, the Mathematical Analysis \& Application Research Fund, and by NSF grants DMS-2347850, and is particularly grateful to recent donors to the Research Fund.  He particularly thanks his coauthors Bogdan Georgiev, Javier G\'omez-Serrano, and Adam Zsolt Wagner for the highly productive and enjoyable collaboration \cite{gdm}, and for generously sharing the outputs of that collaboration for the purposes of writing the current paper.

While some of the results proven here were suggested by the outcome of AI-assisted experiments, the arguments in this paper are completely human-generated.

The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in Theorem~\ref{main-2}. For the three-slope case \eqref{2ab}, the “logarithmic convergence” was “suggested… by experiments… using \texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in \eqref{2ab} rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in \eqref{2ab} using the entropic Pl\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives \eqref{sam}.”

The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in the main theorem. For the three-slope case
\begin{equation}\label{2ab} 2 - \frac{c_2}{D(R;s)} \leq \SD\left(R; s\right) \leq 2 - \frac{c_1}{D(R;s)}
\end{equation}
, the “logarithmic convergence” was “suggested… by experiments… using \texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in the equation above rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in the equation above using the entropic Pl\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives
\begin{equation}\label{sam} 2 - \frac{C_k \log(2+D(R;s))}{D(R;s)} \leq \SD(R; s) \leq 2 - \frac{c_k}{D(R;s)^{k+1}}
    \end{equation}
.”

Question:
Let a slope mean an element of \(\mathbb{Q}\cup\{\infty\}\). For each slope \(r\), define \(\pi_r:\mathbb{Q}^2\to\mathbb{Q}\) by \(\pi_r(x,y)=x+ry\) if \(r\neq\infty\), and \(\pi_\infty(x,y)=y\). For a finite set of slopes \(R\) and a slope \(s\notin R\), define \(\mathrm{SD}(R;s)\) to be the least real number such that
\[
H[\pi_s(X,Y)]\le \mathrm{SD}(R;s)\max_{r\in R} H[\pi_r(X,Y)]
\]
for all finitely supported discrete \(\mathbb{Q}\)-valued random variables \(X,Y\) (not necessarily independent), where \(H[Z]=\sum_z \mathbb{P}(Z=z)\log\frac1{\mathbb{P}(Z=z)}\) is Shannon entropy.

Assume \(R=\{0,1,\infty,r_1,\dots,r_k\}\) is a finite family of slopes of cardinality \(k+3\), and \(s\notin R\). Define the rational complexity \(D=D(R;s)\) to be the least natural number such that
\[
s=\frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
\]
for some polynomials \(P,Q\) of degree at most \(D\) with integer coefficients of magnitude at most \(2^D\), with \(Q(r_1,\dots,r_k)\neq 0\).

Which quantitative estimate holds for \(\mathrm{SD}(R;s)\)?

If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)}.
\]
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_k}{D(R;s)^{k+1}}.
\]

Choices:
B.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1\log(2+D(R;s))}{D(R;s)}.
\]
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_k}{D(R;s)^{k+1}}.
\]
C.
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2.
\]
D.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2}{D(R;s)^2}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)^2}.
\]
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k}{D(R;s)^{k+1}}\le \mathrm{SD}(R;s)\le 2-\frac{c_k\log(2+D(R;s))}{D(R;s)}.
\]
E.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\), independent of \(s\), such that
\[
2-\frac{c_2}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)}.
\]
Moreover, there exist absolute constants \(c,C>0\), independent of \(k\), such that for every finite \(R\) as above,
\[
2-\frac{C\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c}{D(R;s)^{k+1}}.
\]

three-slope logarithmic-free rate vs many-slope logarithmic-loss rate

dropped the nontrivial upper improvement below 2

power of D and side of the estimate carrying the logarithmic loss

dependence of constants on k in the many-slopes bounds