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TRIANGLES CAPTURING MANY LATTICE POINTS

Part of: Additive number theory; partitions Discrete geometry Polytopes and polyhedra Geometry of numbers

Published online by Cambridge University Press:  25 April 2018

Nicholas F. Marshall  and
Stefan Steinerberger
Nicholas F. Marshall
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06511, U.S.A. email nicholas.marshall@yale.edu
Stefan Steinerberger
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06511, U.S.A. email stefan.steinerberger@yale.edu
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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 non-trivial 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 of at most $3/4$ .

MSC classification

Primary: 11P21: Lattice points in specified regions 52B05: Combinatorial properties (number of faces, shortest paths, etc.) 52C05: Lattices and convex bodies in $2$ dimensions
Secondary: 11H06: Lattices and convex bodies 90C27: Combinatorial optimization

Information

Type
Research Article
Information
Mathematika , Volume 64 , Issue 2 , 2018 , pp. 551 - 582
Copyright
Copyright © University College London 2018 

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