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arXiv · 2001.08931

Distribution of missing differences in diffsets

Scott Harvey-Arnold·Steven J. Miller·Fei Peng
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Abstract

Lazarev, Miller and O'Bryant investigated the distribution of $|S+S|$ for $S$ chosen uniformly at random from $\{0, 1, \dots, n-1\}$, 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 shows 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, $P(2n-1-|S-S|=k)$. A corollary of our results is an asymptotic bound for the number of complete rulers of length $n$.

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Scott Harvey-Arnold, Steven J. Miller, Fei Peng. 2020-01-24. Distribution of missing differences in diffsets. https://arxiv.org/abs/2001.08931

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