The supremum of autoconvolutions, with applications to additive number theory

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*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 subset A of {1,2, ..., n}, let g be the maximum multiplicity of any element of the multiset {a+b: a,b in A}. Our main corollary is the inequality gn>0.631|A|^2, which holds uniformly for all g, n, and A.Comment: 17 pages. to appear in IJ

B2[g] Sets and a Conjecture of Schinzel and Schmidt

7 páginas.-- This work was developed during the Doccourse in Additive Combinatorics held in the Centre de Recerca Matemática from January to March 2007.A set of integers A is called a B2[g] set if every integer m has at most g representations of the form m = a + a (prima), with a < = a (prima) and a, a(prima) ∈ A. We obtain a new lower bound for F(g, n), the largest cardinality of a B2[g] set in {1, . . . , n}.More precisely, we prove that lim infn→∞ F(g,n)/√gn <= 2/√π − εg where εg → 0 when g →∞. We show a connection between this problem and another one discussed by Schinzel and Schmidt, which can be considered its continuous version.Both authors are supported by Grants CCG07-UAM/ESP-1814 and DGICYT MTM 2005-04730 (Spain).Peer reviewe

Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics

Barnard and Steinerberger [‘Three convolution inequalities on the real line with connections to additive combinatorics’, Preprint, 2019, arXiv:1903.08731] established the autocorrelation inequality Min_(0≤t≤1)∫_Rf(x)f(x+t) dx ≤ 0.411||f||²L¹, for fϵL¹(R), where the constant 0.4110.411 cannot be replaced by 0.370.37. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics. We show that for f to be extremal for this inequality, we must have max min_(x₁∈R 0≤t≤1)[f(x₁−t)+f(x₁+t)] ≤ min_max(x₂∈ R0≤t≤1)[f(x₂−t)+f(x₂+t)]. Our central technique for deriving this result is local perturbation of f to increase the value of the autocorrelation, while leaving ||f||L¹|| unchanged. These perturbation methods can be extended to examine a more general notion of autocorrelation. Let d, n∈Z⁺, f∈L¹, A be a d×n matrix with real entries and columns a_i for 1≤i≤n and C be a constant. For a broad class of matrices A, we prove necessary conditions for f to extremise autocorrelation inequalities of the form Min_(t∈ [0,1]^d)∫R∏_(i=1)^n f(x+t⋅a_i)dx≤C||f||^nL¹

Generalized Sidon sets

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.Comment: 21 pages, no figure