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 $0.37 < c < 0.411$.

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

Journal of Mathematical Analysis and Applications, 2010arXiv

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Created at: 2019-06-07 18:06:35Z

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