# Triangles capturing many lattice points

## TL;DR

This paper investigates the optimal shape of right triangles with fixed vertices and area to maximize lattice point capture, revealing complex limiting behaviors and fractal structures in the set of optimal slopes.

## Contribution

It proves the existence of a nontrivial limiting set of shapes and identifies a fractal set of slopes with infinitely many optimal triangles for large areas.

## Key findings

- The limiting set of shapes contains infinitely many elements.
- Certain slopes form a fractal set with Minkowski dimension at most 3/4.
- No universal shape is optimal for all large areas.

## Abstract

We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{>0}^2$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed nontrivial and contains infinitely many elements. We also show that there exist `bad' areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3, 3]$ and has Minkowski dimension at most $3/4$.