  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
$\{\alpha\sqrt{n}\}$ is exponential, provided $\alpha^2$ is irrational. For
almost all values of $\alpha$, 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.
