On the number of lattice points in thin sectors
Ezra Waxman, Nadav Yesha
TL;DR
This work counts lattice points in sectors of a circle whose width $2\epsilon$ shrinks as radius $R$ grows, focusing on how the decay rate of $\epsilon$ and the Diophantine type of the slope $\alpha$ affect asymptotics. The authors first show that for slowly shrinking sectors ($\epsilon R\to\infty$) the count is asymptotic to the sector area, extending Gauss-type arguments with precise error $O(R)$. For faster shrinkage they develop a triangle-approximation and leverage good rational approximations to $\alpha$, obtaining sharp asymptotics that differ between rational, irrational finite-type, and Diophantine slopes; in particular, for Diophantine $\alpha$ one gets $S_{\alpha}(\epsilon,R) \sim \text{Area}(\text{Sect}_{\alpha,\epsilon}(R))$ as long as $\epsilon\to0$ with $\epsilon R^{t}\to\infty$ for some $t<2$. In the extreme shrinking regime, the count vanishes for large $R$ when $\epsilon=o(R^{-1-\eta})$ for irrational slopes of finite type $\eta$, highlighting the transition between area-dominated and discrete-lattice-dominated behavior. Overall, the paper provides a comprehensive taxonomy of lattice-point counts in thin sectors across irrational/rational slopes and a range of shrinkage rates, using a blend of geometric, Diophantine, and triangle-approximation techniques.
Abstract
On the circle of radius $R$ centred at the origin, consider a ``thin'' sector about the fixed line $y = αx$ with edges given by the lines $y = (α\pm ε) x$, where $ε= ε_R \rightarrow 0$ as $ R \to \infty $. We establish an asymptotic count for $S_α(ε,R)$, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of $ε$ and on the rationality/irrationality type of $α$. In particular, we demonstrate that if $α$ is Diophantine, then $S_α(ε,R)$ is asymptotic to the area of the sector, so long as $εR^{t} \rightarrow \infty$ for some $ t<2 $.