Related papers: A new expander and improved bounds for $A(A+A)$

For any given set $A$ of nonnegative integers and for any given two positive integers $k_1,k_2$, $R_{k_1,k_2}(A,n)$ is defined as the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In this paper, we prove that if…

In this paper, we study numbers $n$ that can be factored in four different ways as $n = A B = (A + a_1) (B - b_1) = (A + a_2) (B - b_2) = (A + a_3) (B - b_3)$ with $B \le A$, $1 \le a_1 < a_2 < a_3 \le C$ and $1 \le b_1 < b_2 < b_3 \le C$.…

This note proves that there exists positive constants $c_1$ and $c_2$ such that for all finite $A \subset \mathbb R$ with $|A+A| \leq |A|^{1+c_1}$ we have $|AAA| \gg |A|^{2+c_2}$.

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in…

Let A be a finite set of integers and F_A its exponential sum. McGehee, Pigno & Smith and Konyagin have independently proved that the L^1-norm of F_A is at least c log|A| for some absolute constant c. The lower bound has the correct order…

In [Yan22a], we defined so-called ``log-type" GCD sums and proved the lower bounds $\Gamma^{(\ell)}_1(N) \gg_{\ell} \left(\log\log N\right)^{2+2\ell}$. We will establish the upper bounds $\Gamma^{(\ell)}_1(N)\ll_{\ell} \left(\log \log…

We obtain explicit forms of the current best known asymptotic upper bounds for gaps between squarefree integers. In particular we show, for any $x \ge 2$, that every interval of the form $(x, x + 11x^{1/5}\log x]$ contains a squarefree…

We give a simple proof of the Beurling-Malliavin multiplier theorem (BM1) in the particular case of weights that verify the usual finite logarithmic integral condition and such that their log are H{\"o}lder continuous with exponent less…

We prove the following conjecture of Shkredov and Solymosi: every subset $A \subset \mathbf{Z}^2$ such that $\sum_{a\in A\setminus\{0\}} 1/\left\|a\right\|^{2} = +\infty$ contains the three vertices of an isosceles right triangle. To do…

Given a pseudo-effective divisor L we construct the diminished ideal of L, a "continuous" extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier…

We prove a pointwise convergence result for additive ergodic averages associated with certain multiplicative actions of the Gaussian integers. We derive several applications in dynamics and number theory, including: (i) Wirsing's theorem…

Let $r \ge 2$ and $s \ge 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and of the $s$-ary expansion of an irrational real number, viewed as infinite…

A new risk bound is presented for the problem of convex/concave function estimation, using the least squares estimator. The best known risk bound, as had appeared in \citet{GSvex}, scaled like $\log(en) n^{-4/5}$ under the mean squared…

A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i - a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5} |A|.

We obtain improved lower bounds for additive spanners, additive emulators, and diameter-reducing shortcut sets. Spanners and emulators are sparse graphs that approximately preserve the distances of a given graph. A shortcut set is a set of…

We describe in this paper additively left stable sets, i.e. sets satisfying $\left((A+A)-\inf(A)\right)\cap[\inf(A),\sup(A)]=A$ (meaning that $A-\inf(A)$ is stable by addition with itself on its convex hull), when $A$ is a finite subset of…

For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also…

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists).…

We use the theory of cross ratios to construct a real-valued function f of only three variables with the property that for any finite set A of reals, the set f(A) = {f(a,b,c):a,b,c \in A} has cardinality at least C|A|^2/log|A|, for an…

We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on $n$ vertices, with a polynomial number of constraints, requires $\Omega(\sqrt{\sfrac{n}{\log n}})$ many integer variables. By…