Distribution of Missing Differences in Diffsets

Abstract

Lazarev, Miller and O’Bryant [11] investigated the distribution of | S+ S| for S chosen uniformly at random from { 0, 1, ⋯, n- 1 }, and proved the existence of a divot at missing 7 sums (the probability of missing exactly 7 sums is less than missing 6 or missing 8 sums). We study related questions for | S- S|, and show some divots from one end of the probability distribution, P(| S- S| = k), as well as a peak at k= 4 from the other end, P(2 n- 1 - | S- S| = k). A corollary of our results is an asymptotic bound for the number of complete rulers of length n.