Optimal Constants in Two Inequalities (and Additive Combinatorics)

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Posted online: 2019-06-07 18:06:35Z by Stefan Steinerberger47

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 $0.37 < c < 0.411$.

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

    arXiv

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

    Proceedings of the AMS, 2017arXiv

  3. Article Generalized Sidon sets

    Advances in Mathematics, 2010arXiv

  4. Article Improved bounds on the supremum of autoconvolutions

    Journal of Mathematical Analysis and Applications, 2010arXiv


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  • Created at: 2019-06-07 18:06:35Z