Values of random polynomials in shrinking targets

Kelmer, Dubi; Yu, Shucheng

doi:10.1090/tran/8204

Values of random polynomials in shrinking targets
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by Dubi Kelmer and Shucheng Yu PDF

Trans. Amer. Math. Soc. 373 (2020), 8677-8695 Request permission

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 $t^{-\lambda }$, that is valid for almost all indefinite quadratic forms in $n$ variables for any $\lambda <n-2$. 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.

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