A new bound for A(A + A) for large sets

Date/heure
5 janvier 2023
14:30 - 15:30

Oratrice ou orateur
Aliaksei Semchankau

Catégorie d'évènement
Séminaire de Théorie des Nombres de Nancy-Metz

We prove the following structural result, resembling the Arithmetical Regularity Lemma of B. Green, and Graph Container Theorem in hypergraphs:
Lemma: Let $A_1,A_2,\dots ,A_k\subset {\mathbb{F}}_p$ be such that $|A_i|\gg p$ for all $i$. Assume that $(A_1\ast A_2\ast \dots \ast A_k)(a)=o(p^{k-1})$ for some $a\in {\mathbb{F}}_p$.
Then there exist sets $W_1,\dots ,W_k$, which we call wrappers, and sets $Y_1,\dots ,Y_k$, such that:
$(W_1\ast W_2\ast \dots \ast W_k)(b)=o(p^{k-1})$ for some $b\in {\mathbb{F}}_p$ , $A_i\setminus Y_i\subseteq W_i$ and $|Y_i|=o(p)$ for all $i$, $|W_i{|}_{\omega }=p^{o(1)}$ for all $i$, where $|\cdot {|}_{\omega }$ is a Wiener norm.
As a consequence of wrappers having a small Wiener norm, we obtain the following results.
If $A(A+A)$ does not cover all nonzero residues in ${\mathbb{F}}_p$, then $|A|⩽p/8+o(p)$.
If $A$ is both sum-free and satisfies $A=A^{\ast }$, then $|A|⩽p/9+o(p)$.
If $|A|\gg \frac{\log \log p}{\sqrt{\log p}}p$, then $|A+A^{\ast }|⩾(1–o(1))min(2\sqrt{|A|p},p)$.
Constants 1/8, 1/9, and 2 are optimal.
To obtain this result, we use Croot-Laba-Sisask Lemma and properties of Wiener norms.
This continues the work of A. Balog, K. Benjamin, P.-Y. Bienvenu, K. Broughan, F. Hennecart, B. Murphy, M. Rudnev, I. Shkredov, I. Shparlinski, and E. Yazici.