Gaps in the fractional parts of square roots

Abstract

Fractional parts of the first $N$ natural numbers fill the unit interval with asymptotically uniform density. However, the gaps around rational points shrink at an asymptotically lower rate $N^{-1/2}$, and their widths scale with the Thomae ("popcorn") function. This curious connection is derived and related geometrically to shadow pattern in the Euclid's orchard. Generalized cases of higher radicals and their convergence rates, are also investigated.