Raphael Steiner, Université de Bristol, Royaume-Uni

Optimal covering exponent for S³ under twisted Linnik-Selberg-Conjecture

This is a joint work with T. D. Browning and V. Kumaraswamy

Let S³ denote the unit sphere x₁² + x₂² + x₃² + x₄² = 1. In a letter about the efficiency of a universal set of quantum gates, Sarnak raises the question how well points on S³ can be approximated by a rational point of a given small height. Equivalently, given r ∈ N, how small can one take ε such that the union of all ε-balls centred at points x/r ∈ S³ with x ∈ Z⁴ covers the whole of S³. Using Heath-Brown's smooth δ-function variant of the circle method, Sardari is able to show that one can take ε ≫ r^-1/3+δ, for any δ>0. By extending the argument and exploiting further cancellation we are able to show that a twisted version of the Linnik-Selberg-Conjecture on sums of Kloosterman sums implies the optimal choice of ε ≫ r^-1/2+δ. Furthermore we give some progress towards the twisted Linnik-Selberg-Conjecture.