Gap Statistics of the Sequence $\{α\sqrt{n}\}$

The gaps in the sequence $\{\sqrt{n}\}$ were shown by Elkies-McMullen (2004) to have a limiting distribution which is not the exponential distribution. However it is conjectured that the distribution of gaps in the sequence $\{α\sqrt{n}\}$ is exponential, provided $α^2$ is irrational. For almost all values of $α$, we prove an important step in this direction. In particular, we show that all the correlations are Poissonian along a subsequence. Therefore, our result implies that the gap distribution converges to the exponential distribution along the same subsequence.

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