On the Distribution of Rational Squares

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.