On the Distribution of Rational Squares

TL;DR

This paper investigates the properties and behavior of the functions  and , which relate to the placement of integer squares within scaled intervals, revealing their complex and chaotic nature.

Contribution

It introduces new bounds and criteria for the functions  and , enhancing understanding of their distribution and behavior.

Findings

 exhibits chaotic behavior.

Derived bounds for  are sharp under certain conditions.

Criteria for the bounds' sharpness are established.

Abstract

Let $a$ be a positive integer, and let $\sigma(a)$ denote the least natural number $s$ such that an integer square lies between $s^2 a$ and $s^2 (a+1)$; let $\tau_s(a)$ denote the number of such integer squares. The function $\sigma(a)$ and the sequence $(\tau_s(a))_{s \in \mathbb{Z}^+}$ are studied, and are observed to exhibit surprisingly chaotic behavior. Upper- and lower-bounds for $\sigma(a)$ are derived, as are criteria for when they are sharp.