For positive integers and , let denote the number of points such that . In other words, is the number of lattice points in an -dimensional -ball of radius . Bump, Choi, Kurlberg and Vaaler have observed that the function enjoys the following symmetry
This is to say that the -dimensional -ball of radius contains exactly the same number of lattice points as the -dimensional -ball of radius . The original proof follows from establishing the generating function identity
and noting that it is symmetric in and . Some time ago Jeff Vaaler mentioned to me that it would be nice to find a bijective proof of this fact. In this post I will give a simple bijective proof that Allison Bishop and I found.