On the modular sumset partition problem

Abstract

A sequence m1≥m2≥⋯≥mk of k positive integers isn-realizable if there is a partition X1,X2,…,Xk of the integer interval [1,n] such that the sum of the elements in Xi is mi for each i=1,2,…,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k≤(p−1)/2 are realizable for any prime p≥3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n≥(4k)3, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.