Let $X_1,X_2,\ldots$ be i.i.d. finitely supported random variables in a torsion-free abelian group, and write $S_k=X_1+\cdots+X_k$, and $H(S_k)$ is the Shannon entropy $S_k$, for all $k \ge 1$. We prove that, for every fixed $n\geq1$, \[ H(S_{n+1})-H(S_n) \geq \frac12\log\frac{n+1}{n} -o_{H(X_1)\to\infty}(1), \] uniformly over the ambient group and the input law. This proves a conjecture of Tao [29] in 2010.
For $P=X_1^2+\cdots + X_n^2$, let $T_t^Pf(x)$ denote the solution to the linear Schr\"{o}dinger equation at time $t$. In 1980, Carleson asked for the minimal regularity of an initial data function $f\in H^s(\mathbb{R}^n)$ that guarantees pointwise convergence of $T_t^Pf(x)$ to $f(x)$ as $t\rightarrow 0$. This was resol...
We prove the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in $[0,1]$, and a unique optimal arm. For each suboptimal arm $i$, let $\Delta_i=\mu_*-\mu_i$ be its gap from the optimal mean, and write $H=\sum_{i\ne *}\Delta_i^{-2}$. Let $p_r$ be the...
P. M. Aronow, Nathan Kallus, Patrick Lopatto· 1 citation
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sa...
Asaf Cohen Antonir, Ilay Hoshen, M. Zhukovskii· 1 citation
For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusi...
Omkar Baraskar, Prashant Gokhale, Sarvagya Jain et al.· 0 citations
Suppose $X,Y$ are independent random variables with values in a compact abelian group $(G,+)$. We examine the following two entropy power-type inequalities: $h(X+Y)\geq \frac{1}{2}h(X)+\frac{1}{2}h(Y)$ and $h(X+Y)\geq \max\{h(X),h(Y)\}$, where the entropy $h(Z)$ of a $G$-valued random variable $Z$ is defined in terms o...
Lampros Gavalakis, I. Kontoyiannis, Sharang M. Sriramu et al.· 1 citation
We prove the Zygmund conjecture in all parameters: the maximal operator associated with products of cubes with side lengths $(s_1,\ldots,s_{m-1},\phi(s_1,\ldots,s_{m-1}))$, where $\phi$ is any positive coordinatewise nondecreasing function, satisfies the weak $L(\log L)^{m-2}$ estimate for every $m\geq3$. The estimate...
Henri Martikainen· 0 citations
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