We obtain a substantially improved lower bound for the minimum overlap problem asked by Erdős. Our approach uses elementary Fourier analysis to translate the problem to a convex optimization program.

Abstract Given any positive integer n , consider partitions of the integers 1, 2, …, 2 n into two disjoint classes { a i } and { b i } with n elements in each class. Denote by M k the number of… Expand

The topics covered are: additive representation functions, the Erdős-Fuchs theorem, multiplicative problems (involving general sequences), additive and multiplicative Sidon sets, hybrid problems (i.e., problems involving both special and general sequences, arithmetic functions and the greatest prime factor func- tion and mixed problems.Expand

For a given partition of (1, 2, ..., 2n) into two disjoint subsets A and B with n elements in each, consider the maximum number of times any integer occurs as the difference between an element of A… Expand

A significant special case of the problems which could be solved were those whose constraints were given by semidefinite cones, and these have a wide range of applications, some of which are discussed in Section 5, and can still be solved efficiently using interior point methods.Expand

Five summation methods and their variations are analyzed here and no one method is uniformly more accurate than the others, but some guidelines are given on the choice of method in particular cases.Expand