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arXiv · 1210.6429

Multiplicative Congruences with Variables from Short Intervals

Jean Bourgain·Moubariz Z. Garaev·Sergei V. Konyagin·Igor E. Shparlinski
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Abstract

Recently, several bounds have been obtained on the number of solutions to congruences of the type $$ (x_1+s)...(x_ν+s)\equiv (y_1+s)...(y_ν+s)\not\equiv0 \pmod p $$ modulo a prime $p$ with variables from some short intervals. Here, for almost all $p$ and all $s$ and also for a fixed $p$ and almost all $s$, we derive stronger bounds. We also use similar ideas to show that for almost all primes, one can always find an element of a large order in any rather short interval.

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Jean Bourgain, Moubariz Z. Garaev, Sergei V. Konyagin, Igor E. Shparlinski. 2012-10-24. Multiplicative Congruences with Variables from Short Intervals. https://arxiv.org/abs/1210.6429

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