On the modular sumset partition problem

A sequence m 1 ≥ m 2 ≥ ⋯ ≥ m k of k positive integers isn-realizable if there is a partition X 1 , X 2 , … , X k of the integer interval [ 1 , n ] such that the sum of the elements in X i is m i 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 / p Z 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 ≥ ( 4 k ) 3 , an n -feasible sequence of length k is n -realizable if and only if it does not contain forbidden subsequences of elements smaller than n , a natural obstruction for n -realizability.