Linear Combinations of Sets of Consecutive Integers

Let k-l,ml,..~+k denote non-negative integers, and suppose the greatest common divisor of ml,...,mk is 1 . We show that if '1, "',sk are sufficiently long blocks of consecutive integers, then the set mlSl+ . ..+mkSk contains a sizable block of consecutive integers. For example; if m and n are relatively prime natural numbers, and U, U ? Vt V are integers with U-u 2 n-l , V-v 2 m-l 1 then the set m{u,u+l, . . ..U)+n[v.v+l,...,V] contains the set cmu+nva(m,n), . . . ,mU+nV-rr(m,n)] where o(m,n) = (m-l)(n-1) is the largest number such that a(m,n)-1 cannot be expressed in the form mx+ny with x and y non-negative integers. This research was supported in part by the Office of Naval Research under contract number N-OOO&67-A-0112-0057 NR 044-402, and by the National Science Foundation under grant number GJ-992. Reproduction in whole or in part is permitted for any purpose of the United States Government. -/ * Currently Visiting Professor, Faculty of Mathematics, Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada. d