Distribution of the integral points on quadrics

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.



University of Wisconsin Madison


Naser Talebi Zadeh Sardari