# Values of random polynomials in shrinking targets

## TL;DR

This paper establishes asymptotic formulas for counting integer vectors with polynomial values in shrinking targets, confirming predictions for indefinite quadratic forms and extending results to higher degree polynomials.

## Contribution

It provides effective asymptotic formulas for integer solutions in shrinking targets for indefinite quadratic forms and higher degree polynomials, confirming prior predictions.

## Key findings

- Asymptotic formulas valid for almost all indefinite quadratic forms.
- Existence of integer solutions in shrinking targets for these forms.
- Extension of results to higher degree random polynomials.

## 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.