Gap widths in the fractional parts of roots

The fractional parts of roots of natural numbers, [Formula: see text] fill the unit interval with asymptotically uniform density. They also have other interesting statistical properties, such as the asymptotic distribution of gaps and their Poissonian pair correlations. In this paper, we show that the gaps around rational points shrink at an asymptotically lower rate [Formula: see text], where N is the length of the sequence. We derive an explicit expression for the gap width at each rational point, which scales as Thomae’s (“popcorn”) function. Our result is not limited to the case of square roots but also covers gap widths for certain subsequences. Finally, we show that the gap width shrinks faster than [Formula: see text] at all irrational points. While our arguments transfer directly to higher radicals for rational points, we discuss how the behavior of gaps around irrational points is closely connected to questions from Diophantine approximation and in particular the irrationality measure.