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Counting lattice points inside a ball of large radius in Euclidean space is aclassical problem in analytic number theory, dating back to Gauss. We propose avariation on this problem: studying the asymptotics of the measure of aninteger lattice of affine planes inside a ball. The first term is the volume ofthe ball; we study the size of the remainder term. While the classical problemis equivalent to counting eigenvalues of the Laplace operator on the torus, ourvariation corresponds to the integrated density of states of the Laplaceoperator on the product of a torus with Euclidean space. The asymptotics weobtain are then used to compute the density of states of the magneticSchroedinger operator.

A generalised Gauss circle problem and integrated density of states