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

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