# Optimal Constants in Two Inequalities (and Additive Combinatorics)

Unconfirmed

Posted online: 2019-06-07 18:06:35Z by Stefan Steinerberger23

Cite as: P-190607.1

• Classical Analysis and ODEs
• Combinatorics
• Number Theory
• Probability

### Problem's Description

This is a classical problem in additive combinatorics that can be equivalently phrased as a problem in real analysis (this equivalence is due to Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa): what random variable $X$ with compact support on $\mathbb{R}$ has the property that the distribution of $X+X$ is as flat as possible?

More precisely, let the function $f:\mathbb{R} \rightarrow \mathbb{R}$ be nonnegative and compactly supported in $(-0.25, 0,25)$. What is the best constant $c$ in the inequality $$\max_{t \in \mathbb{R}} \int_{\mathbb{R}}{f(t-x)f(x) dx} \geq c \left( \int_{\mathbb{R}}^{}{ f(x)dx} \right)^2.$$ It is known that the optimal constant satisfies $$1.28 \leq c \leq 1.52$$ where the lower bound is due to Cloninger & Steinerberger and the upper bound is due to Matolcsi & Vinuesa. 

There is a dual question that also assumes the form of a very simple inequality on $\mathbb{R}$. Let $f \in L^1(\mathbb{R})$, what is the best constant $c>0$ in the inequality $$\min_{0 \leq t \leq 1} \int_{\mathbb{R}}{f(x) f(x+t) dx} \leq c \| f\|_{L^1(\mathbb{R})}^2$$ It is conjectured that this is related to the notion of g-difference bases in additive combinatorics. The best bounds are $0.37 < c < 0.411$.

arXiv

2. ## Article On suprema of autoconvolutions with an application to Sidon sets

Proceedings of the AMS, 2017arXiv