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Density of non-residues in Burgess-type intervals and applications

W. D. Banks·M. Z. Garaev·D. R. Heath-Brown·I. E. Shparlinski
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Abstract

We show that for any fixed $\eps>0$, there are numbers $δ>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $δN$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.

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W. D. Banks, M. Z. Garaev, D. R. Heath-Brown, I. E. Shparlinski. 2007-09-25. Density of non-residues in Burgess-type intervals and applications. https://doi.org/10.1112/blms%2Fbdm111

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