Related papers: Some optimal links between generations of correlat…

We construct real numbers $\alpha$ for which the pair correlation function \[N^{-1}#\{m<n\le N:||\alpha m^2-\alpha n^2||\le XN^{-1}\}\] tends to $X$ as $N$ grows. Moreover we show for any "Diophantine" $\alpha$ that the pair correlation…

The main results of this paper concern growth in sums of a $k$-convex function $f$. Firstly, we streamline the proof of a growth result for $f(A)$ where $A$ has small additive doubling, and improve the bound by removing logarithmic factors.…

We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…

We review a result obtained with Andrew Ledoan and Marco Merkli. Consider a random analytic function $f(z) = \sum_{n=0}^{\infty} a_n X_n z^n$, where the $X_n$'s are i.i.d., complex valued random variables with mean zero and unit variance,…

We give universal bounds on the fraction of nontrivial zeros having given multiplicity for L-functions attached to a cuspidal automorphic representation of $\mathrm{GL}_m/\mathbb{Q}$. For this, we apply the higher-level correlation…

Functionality ($\mathrm{fun}$) is a graph parameter that generalizes graph degeneracy defined by Alecu et al. [JCTB, 2021]. They research the relation of functionality to many other graphs parameters (tree-width, clique-width, VC-dimension,…

We consider the symmetric FEM-BEM coupling that connects two linear elliptic second order partial differential equations posed in a bounded domain $\Omega$ and its complement, where the exterior problem is restated by an integral equation…

We study near-extremal n-point correlation functions of chiral primary operators, in which the maximal scale dimension k is related to the others by k=\sum_i k_i-m with m equal to or smaller than n-3. Through order g^2 in field theory, we…

We show the pointwise convergence of the averages \[ \mathcal{A}_N f(x) = \frac{1}{\# \mathbf{B}_N} \sum_{n \in \mathbf{B}_N} f(x + n) \] for $f \in \ell^1(\mathbb{Z})$ where $\mathbf{B}_N = \mathbf{B} \cap [1, N]$, and $\mathbf{B}$ is a…

Let $f:\mathbb{C}^2\to\mathbb{C}$ be an inner non-degenerate mixed polynomial with a nice Newton boundary with $N$ compact 1-faces. In the first part of this series of papers we showed that $f$ has a weakly isolated singularity and that its…

We extend the Matom\"{a}ki-Radziwi\l\l{} theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a…

We consider extremal values of sum-Balaban index among graphs on $n$ vertices. We determine that the upper bound for the minimum value of the sum-Balaban index is at most $4.47934$ when $n$ goes to infinity. For small values of $n$ we…

Let $X_1, X_2,\dots$ be a short-memory linear process of random variables. For $1\leq q<2$, let $\cF$ be a bounded set of real-valued functions on $[0,1]$ with finite $q$-variation. It is proved that…

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set $[N]=\{1,2,\hdots,N\}$ was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph…

Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to \emph{negative correlation} properties. However, what if an application naturally calls for dependent…

Inspired by a question asked on the list {\tt mathfun}, we revisit {\em Kempner-like series}, i.e., harmonic sums $\sum' 1/n$ where the integers $n$ in the summation have ``restricted'' digits. First we give a short proof that $\lim_{k \to…

We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a…

We prove new bounds on the average sensitivity of the indicator function of an intersection of $k$ halfspaces. In particular, we prove the optimal bound of $O(\sqrt{n\log(k)})$. This generalizes a result of Nazarov, who proved the analogous…

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M…

This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…