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  • Nicholas F. Marshall (a1) and Stefan Steinerberger (a2)


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 $3/4$ .



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1. Antunes, P. R. S. and Freitas, P., Optimal spectral rectangles and lattice ellipses. Proc. R. Soc. Lond. Ser. A 469(2150) 2012.
2. Antunes, P. R. S. and Freitas, P., Optimisation of eigenvalues of the Dirichlet Laplacian with a surface area restriction. Appl. Math. Optim. 73(2) 2016, 313328.
3. Ariturk, S. and Laugesen, R. S., Optimal stretching for lattice points under convex curves. Port. Math. 74(2) 2017, 91114.
4. Beck, M. and Robins, S., Computing the Continuous Discretely (Undergraduate Texts in Mathematics), Springer (New York, 2007).
5. Berger, A., The eigenvalues of the Laplacian with Dirichlet boundary condition in ℝ2 are almost never minimized by disks. Ann. Global Anal. Geom. 47(3) 2015, 285304.
6. Besicovitch, A. S., On the linear independence of fractional powers of integers. J. Lond. Math. Soc. (2) 15 1940, 36.
7. Boreico, I., Linear independence of radicals. The Harvard College Mathematics Review 2(1) 2008, 8792.
8. Bucur, D. and Freitas, P., Asymptotic behaviour of optimal spectral planar domains with fixed perimeter. J. Math. Phys. 54(5) 2013,053504.
9. Kronecker, L., Näherungsweise ganzzahlige Auflösung linearer Gleichungen. Berl. Ber. 1179–1193 1884, 12711299.
10. Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences (Pure and Applied Mathematics), Wiley-Interscience (1974).
11. Laugesen, R. and Liu, S., Optimal stretching for lattice points and eigenvalues. Ark. Mat. (to appear).
12. Pick, G., Geometrisches zur Zahlenlehre. Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen Lotos in Prag 19 1899, 311319.
13. van den Berg, M., Bucur, D. and Gittins, K., Maximising Neumann eigenvalues on rectangles. Bull. Lond. Math. Soc. 48(5) 2016, 877894.
14. van den Berg, M. and Gittins, K., Minimising Dirichlet eigenvalues on cuboids of unit measure. Mathematika 63(2) 2017, 469482.
15. Weyl, H., Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77(3) 1916, 313352.
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  • Nicholas F. Marshall (a1) and Stefan Steinerberger (a2)


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