The supremum of autoconvolutions, with applications to additive number theory
Martin, Greg ; O’Bryant, Kevin
Illinois J. Math., Tome 53 (2009) no. 1, p. 219-235 / Harvested from Project Euclid
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We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f∈L1(ℝ)∩L2(ℝ) is nonnegative and supported on an interval of length I, then the supremum of f*f is at least 0.631‖f‖12/I. This improves the previous bound of 0.591389‖f‖12/I. Consequently, we improve the known bounds on several related number-theoretic problems. For a set A⊆{1, 2, …, n}, let g be the maximum multiplicity of any element of the multiset {a1+a2 : ai∈A}. Our main corollary is the inequality gn>0.631|A|2, which holds uniformly for all g, n, and A.
Publié le : 2009-05-15
Classification:  42A85,  11P70,  11B83 
@article{1264170847,
     author = {Martin, Greg and O'Bryant, Kevin},
     title = {The supremum of autoconvolutions, with applications to additive number theory},
     journal = {Illinois J. Math.},
     volume = {53},
     number = {1},
     year = {2009},
     pages = { 219-235},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1264170847}
}

Martin, Greg; O’Bryant, Kevin. The supremum of autoconvolutions, with applications to additive number theory. Illinois J. Math., Tome 53 (2009) no. 1, pp.  219-235. http://gdmltest.u-ga.fr/item/1264170847/