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Spring 2009 The supremum of autoconvolutions, with applications to additive number theory
Greg Martin, Kevin O’Bryant
Illinois J. Math. 53(1): 219-235 (Spring 2009). DOI: 10.1215/ijm/1264170847


We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if $f\in L^1({\mathbb R}) \cap L^2({\mathbb R})$ is nonnegative and supported on an interval of length $I$, then the supremum of $f\ast f$ is at least $0.631 \|f\|_1^2/I$. This improves the previous bound of $0.591389 \|f\|_1^2/I$. Consequently, we improve the known bounds on several related number-theoretic problems. For a set $A\subseteq\{1,2,\dots,n\}$, let $g$ be the maximum multiplicity of any element of the multiset $\{a_1+a_2 : a_i\in A\}$. Our main corollary is the inequality $g n > 0.631 |A|^2$, which holds uniformly for all $g$, $n$, and $A$.


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Greg Martin. Kevin O’Bryant. "The supremum of autoconvolutions, with applications to additive number theory." Illinois J. Math. 53 (1) 219 - 235, Spring 2009.


Published: Spring 2009
First available in Project Euclid: 22 January 2010

zbMATH: 1196.42008
MathSciNet: MR2584943
Digital Object Identifier: 10.1215/ijm/1264170847

Primary: 11B83 , 11P70 , 42A85‎

Rights: Copyright © 2009 University of Illinois at Urbana-Champaign


Vol.53 • No. 1 • Spring 2009
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