Distribution of the integral points on quadrics

Distribution of the integral points on quadrics -Naser Talebi Zadeh Sardari

Naser Talebi Zadeh Sardari
University of Wisconsin Madison
January 9, 2019
Motivated by questions in computer science, we consider the problem of approximating local points (real or p-adic points) on the unit sphere S^d optimally by the projection of the integral points lying on R*S^d, where R^2 is an integer. We present our numerical results which show the diophantine exponent of local point on the sphere is inside the interval [1, 2-2/d]. By using the Kloosterman's circle method, we show that the diophantine exponent is less than 2-2/d for every d>3. By using the theta-lift and Ramanujan bound on the Fourier coefficients of the holomorphic modular forms we prove that the diophantine exponent is 1+o(1) for almost all local points and odd d>=3 and even d>=2 by assuming R is an integer. This generalizes the result of Sarnak for d=3 to higher dimensions.