Gaps in the fractional parts of square roots

2020/12/27 by Simon Čopar, Čopar, Simon
Computer Science · Mathematics · #11B05 #11K06 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2012.14019

openalex publication_date 2020/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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