Type: Article

On the distribution of modular inverses from short intervals

Garaev, Moubariz Z.; Shparlinski, Igor E.;

Abstract:

The authors study the irregularities of distribution of sequences of modular inverses. Given a prime p and a positive integer N<p such sequences are defined as x¯¯¯,x=1,…,N, based on the conditions x¯¯¯x?1(modp)and1?x¯¯¯<p. The authors are interested in the largest possible values of N for which such a sequence is not uniformly distributed. The authors apply an interesting technique based on smooth numbers to generate infinitely many primes p such that its successor, p+1, has many divisors in the interval [N/2,N]. This construction reveals irregularities in the sequences of inverses which allow the authors to derive their main results. In particular, it is shown in Theorem 1.1 that nonuniform distribution can occur in segments of sequences of inverses which are longer than one would expect. In Theorem 1.3, the authors show that segments of length N of a sequence of inverses may have a poorer distribution than intuitively expected. Both results are accompanied by lower bounds on related exponential sums.