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

## TL;DR

This paper investigates three convolution inequalities on the real line, connecting them to additive combinatorics, and provides bounds and conjectures related to g-Sidon sets and difference bases.

## Contribution

It proves new convolution inequalities related to additive combinatorics, establishes bounds on integral expressions, and proposes conjectures on the structure of difference bases.

## Key findings

- Established a lower bound for a convolution maximum related to g-Sidon sets.
- Derived an upper bound for the minimum of a convolution integral related to difference bases.
- Presented a combined integral inequality involving L^1 and L^2 norms.

## Abstract

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f \in L^1(-1/4, 1/4)$, $$ \max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}}{f(t-x) f(x) dx} \geq 1.28 \left( \int_{-1/4}^{1/4}{f(x) dx}\right)^2$$ 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 $f \in L^1(\mathbb{R})$, $$ \min_{0 \leq t \leq 1}\int_{\mathbb{R}}{f(x) f(x+t) dx} \leq 0.42 \|f\|_{L^1}^2,$$ 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 $f \in L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, $$ \int_{-\frac{1}{2}}^{\frac{1}{2}}{ \int_{\mathbb{R}}{f(x) f(x+t) dx}dt} \leq 0.91 \|f\|_{L^1}\|f\|_{L^2}$$