Values of random polynomials in shrinking targets

Abstract

Relying on the classical second moment formula of Rogers we give an effective asymptotic formula for the number of integer vectors $v$ in a ball of radius $t$, with value $Q(v)$ in a shrinking interval of size $t^{-\kappa }$, that is valid for almost all indefinite quadratic forms in $n$ variables for any $\kappa <n-2$. This implies in particular, the existence of such integer solutions establishing the prediction made by Ghosh Gorodnik and Nevo. We also obtain similar results for random polynomials of higher degree.

Cited by in corpus (1)

• Some new results in quantitative Diophantine approximation