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- W2922558231 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 $f in L^1(mathbb{R}) cap L^2(mathbb{R})$, $$ int_{-frac{1}{2}}^{frac{1}{2}}{ int_{mathbb{R}}{f(x) f(x+t) dx}dt} leq 0.91 |f|_{L^1}|f|_{L^2}$$" @default.
- W2922558231 created "2019-04-01" @default.
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- W2922558231 date "2019-03-20" @default.
- W2922558231 modified "2023-09-27" @default.
- W2922558231 title "Three Convolution Inequalities on the Real Line with Connections to Additive Combinatorics." @default.
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