Lattice Points in High-Dimensional Spheres

The Gauss Circle Problem is a classic open problem in number theory concerning the number of lattice points contained in a large circle.

Optimal error bounds are known in these approximations in a generalization of the Gauss Circle Problem to spheres in dimensions four and above.

In this post, I’ll give a purely analytic proof of this result for even dimensions greater than four, and explain why the method fails in the other cases.

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Sums of Squares and Density

Classification theorems of Euler, Lagrange, and Legendre describe the sets of integers that can be written as the sum of 2, 3, and 4 squares. In the last two cases, it follows easily that the density of these sets are 5/6 and 1.

The question of density is not so simple in the case of two squares. In this post, we resolve using an unexpected tool — Dirichlet’s theorem on primes in arithmetic progressions.

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Maximal Products of a Given Sum

Back in high school, I came across the following contest problem – “What is the largest product of a set of positive integers totaling 20?”

It’s a fun problem, so don’t rush past the spoiler tags too fast. In this post, we’ll give the solution to this problem and discuss a “continuous” version of this question. Namely, what happens when we’re allowed to include real numbers in our product?

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