Three convolution inequalities on the real line with connections to additive combinatorics

We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative f is an element of L-1 (-1/4, 1/4), max(-1/2 <= t <= 1/2) integral(R) f(t - x) f(x)dx >= 1.28 (integral(1/4)(-1/4) f(x)dx)(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 is an element of L-1 (R), min(0 <= t <= 1) integral(R) f(x)f(x + t)dx <= 0.42 parallel to f parallel to(2)(L1), 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 is an element of L-1(R) boolean AND L-2(R), integral(1/2)(-1/2) integral(R) f(x)f(x + t)dxdt <= 0.91 parallel to f parallel to(L1)parallel to f parallel to(L2). (C) 2019 Elsevier Inc. All rights reserved.