Optimal Constants in Two Inequalities (and Additive Combinatorics)

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 $0.37 < c < 0.411$.

1. Article Three Convolution Inequalities on the Real Line with Connections to Additive Combinatorics

   • Richard Barnard,
   • Stefan Steinerberger

   arXiv

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

   • Alexander Cloninger,
   • Stefan Steinerberger

   Proceedings of the AMS, 2017arXiv

3. Article Generalized Sidon sets

   • J. Cilleruelo,
   • I. Ruzsa,
   • C. Vinuesa

   Advances in Mathematics, 2010arXiv

4. Article Improved bounds on the supremum of autoconvolutions

   • M. Matolcsi,
   • C. Vinuesa

   Journal of Mathematical Analysis and Applications, 2010arXiv