Multiplicative Congruences with Variables from Short Intervals
Jean Bourgain, Moubariz Z. Garaev, Sergei V.Konyagin, Igor E., Shparlinski

TL;DR
This paper improves bounds on the number of solutions to certain multiplicative congruences with variables from short intervals and demonstrates the existence of elements with large order in short intervals for almost all primes.
Contribution
It provides stronger bounds for solutions to multiplicative congruences and shows that for almost all primes, large order elements can be found in short intervals.
Findings
Stronger bounds on solutions to multiplicative congruences.
Existence of large order elements in short intervals for almost all primes.
Results hold for almost all primes and fixed parameters.
Abstract
Recently, several bounds have been obtained on the number of solutions to congruences of the type modulo a prime with variables from some short intervals. Here, for almost all and all and also for a fixed and almost all , 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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Taxonomy
TopicsAnalytic Number Theory Research · Limits and Structures in Graph Theory · Finite Group Theory Research
