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

Abstract

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f\in L^1(-1/4,1/4)$, $$\underset{-1/2\le t\le 1/2}{\max }{\int }_{\mathbb{R}}f(t-x)f(x)dx\ge 1.28{\left({\int }_{-1/4}^{1/4}f(x)dx\right)}^2$$ which is related to $g-$Sidon sets (1.28 cannot be replaced by 1.52). We prove a dual statement, related to difference bases, and show that for $f\in L^1(\mathbb{R})$, $$\underset{0\le t\le 1}{\min }{\int }_{\mathbb{R}}f(x)f(x+t)dx\le 0.42\Vert f{\Vert }_{L^1}^2,$$ where the constant 1/2 is trivial, 0.42 cannot be replaced by 0.37. This suggests a natural conjecture about the asymptotic structure of $g-$difference bases. Finally, we show for all functions $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, $${\int }_{-\frac{1}{2}}^{\frac{1}{2}}{\int }_{\mathbb{R}}f(x)f(x+t)dxdt\le 0.91\Vert f{\Vert }_{L^1}\Vert f{\Vert }_{L^2}$$