Related papers: Variations on the Sum-Product Problem

We make progress on two interrelated problems at the intersection of geometric measure theory, additive combinatorics and harmonic analysis: the discretised sum-product problem, and the dimension of Furstenberg sets. Along the way, we…

Many identities written by $P=S=C$ are obtained, where $P$ infinite products, $S$ infinite series, and $C$ continued fractions. Such equality is called {\it triplicity}, and it can be used to compute the values of infinite series. It is…

Let $A$ be an infinite set of natural numbers. For $n\in \mathbb{N}$, let $r(A, n)$ denote the number of solutions of the equation $n=a+b$ with $a, b\in A, a\le b$. Let $|A(x)|$ be the number of integers in $A$ which are less than or equal…

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this paper, we prove that if $\sigma_1(G)<2+\frac{11}{|G|}$\,, then $G$ is supersolvable. In particular, some new characterizations of the well-known groups…

We study several integrals that contain the infinite product ${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$ in the denominator of their integrand. These considerations lead to closed form evaluation…

A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Bedert and Kravitz proved that this statement holds whenever $|A| \leq e^{c(\log…

Let $A$ be a subset of the cyclic group $\mathbf{Z}/p\mathbf{Z}$ with $p$ prime. It is a well-studied problem to determine how small $|A|$ can be if there is no unique sum in $A+A$, meaning that for every two elements $a_1,a_2\in A$, there…

For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>=…

Given a finite set of integers $A$, its sumset is $A+A:= \{a_i+a_j \mid a_i,a_j\in A\}$. We examine $|A+A|$ as a random variable, where $A\subset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$…

We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems.…

In this paper we study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not=2$, where ${\rm ch}(F)$ is the characteristic of $F$. For any integer $k\ge4$, we show that each $x\in F$ can be written as $a_1+\ldots+a_k$ with…

We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems…

We systematically explore a class of constrained optimization problems with linear objective function and constraints that are linear combinations of logarithms of the optimization variables. Such problems can be viewed as a generalization…

We prove that if $A\subseteq \{1,\dots,N\}$ does not contain any non-trivial three-term arithmetic progression, then $$|A|\ll \frac{(\log\log N)^{3+o(1)}}{\log N}N\,.$$

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is…

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+\omega(N)$ for some increasing unbounded function $\omega$. The question is generally…

We present a new advancement in the sum and difference of sets problem, which improves upon recent results by both DeepMind's AlphaEvolve ($\theta = 1.1584$) and subsequent explicit constructions ($\theta = 1.173050$). In this work, we…

The sum-product phenomena over a finite extension K of $\mathbb{Q}_p$ is explored. The main feature of the results is the fact that the implied constants are independent of $p$.

Let $A$ be a subset of integers and let $2\cdot A+k\cdot A=\{2a_1+ka_2 : a_1,a_2\in A\}$. Y. O. Hamidoune and J. Ru\' e proved that if $k$ is an odd prime and $A$ a finite set of integers such that $|A|>8k^k$, then $|2\cdot A+k\cdot A|\ge…