Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average

Abstract

We show that, for sufficiently large integers m and X, for almost all a = 1, . . . , m the ratios a/x and the products ax, where |x| X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums K r,s (p) = p−1 n=1 exp(2πi(rn + sn −1 )/p), for primes p T is in accordance with the Sato-Tate conjecture.

Learning Sums of Independent Random Variables with Sparse Collective Support

“…This in turn suggests that by drawing a large random set of values for a 2 , it should be possible to obtain a large family of distributions of the form (a 2 ·U r mod a 3 ) such that any two of them have large variation distance. We make this intuition precise using a number-theoretic equidistribution result of Shparlinski [Shp08] and a probabilistic argument showing that indeed a random set of (a 3 ) 1/3 choices of a 2 is likely to have the desired property. This gives a "hard family" of size (a 3 ) 1/3 , leading to an Ω(log a 3 ) = Ω(log a max ) lower bound via Fano's inequality.…”

The equation x1x2=x3x4+λ in fields of prime order and applications

“…For further information on this subject we refer the reader to [9], to the works of Shparlinski [13,14], and therein references.…”

“…Distribution properties of its solutions are proved to be important in many applications, see, for example, [1,5,7,9,13,14].…”

Distribution of Farey fractions in residue classes and Lang–Trotter conjectures on average

“…We use some results of [20] to show that the elements of this set are uniformly distributed in residue classes modulo a prime p. More precisely, we prove: (1) .…”