Effective Counting and Spiralling of Lattice Approximates

Abstract: We will prove an effective version of Dirichlet’s approximation theorem, giving the error between the number of rational approximations to a real vector with denominator less than some real number T and the asymptotic growth of this count. Additional results for linear forms can be obtained, as well as results measuring the direction of these approximates, known as ‘spiralling of lattice approximates’. These results are obtained by reformulating the number-theoretic problem to the context of homogeneous spaces of unimodular lattices. The advantage of this reformulation is that we have more tools to deal with the problem, such as Siegel’s mean value theorem and Rogers’ higher moment formula. The proof involves using the ergodic properties of diagonal flows on this homogeneous space to calculate the number of lattice approximates, bounding the second moment of the count, then applying an effective ergodic theorem due to Gaposhkin. Particular attention is paid to the case of primitive lattices in two-dimensions, where Rogers’ theorem fails. In this case, we apply a new theorem by Kleinbock and Yu to obtain a better error term than previous results due to Schmidt.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar