A. Walker

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20182020

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6 papers · 1 filter

math.NT2020

Extremal problems for GCDs

Ben Green, Aled Walker

We prove that if $A \subseteq [X, 2 X]$ and $B \subseteq [Y, 2 Y]$ are sets of integers such that $g cd (a, b) \geq D$ for at least $δ ∣ A ∣∣ B ∣$ pairs $(a, b) \in A \times B$ then $|A||B|…

math.NT2020

A tight structure theorem for sumsets

Andrew Granville, Aled Walker

Let $A = {0 = a_{0} < a_{1} < \dots < a_{ℓ + 1} = b}$ be a finite set of non-negative integers. We prove that the sumset $N A$ has a certain easily-described structure, provided t…

math.NT2020

On the triple correlations of fractional parts of $n^{2} α$

Niclas Technau, Aled Walker

For fixed $α \in [0, 1]$ , consider the set $S_{α, N}$ of dilated squares $α, 4 α, 9 α, \dots, N^{2} α$ modulo $1$ . Rudnick and Sarnak conjectured that for Lebesgue almost all such $α$ t…

math.NT2019

Linear inequalities in primes

Aled Walker

In this paper we prove an asymptotic formula for the number of solutions in prime numbers to systems of simultaneous linear inequalities with algebraic coefficients. For $m$ simult…

math.NT2018

Arithmetic Progressions with Restricted Digits

Aled Walker, Alexander Walker

For an integer $b ⩾ 2$ and a set $S \subset {0, \dots, b - 1}$ , we define the Kempner set $K (S, b)$ to be the set of all non-negative integers whose base- $b$ digital…

math.NT2018

GCD sums and sum-product estimates

Thomas F. Bloom, Aled Walker

In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretè…

Extremal problems for GCDs

A tight structure theorem for sumsets

On the triple correlations of fractional parts of n2α

Linear inequalities in primes

Arithmetic Progressions with Restricted Digits

GCD sums and sum-product estimates

On the triple correlations of fractional parts of $n^{2} α$