Multiplicative translates of subgroups in residue rings
We will study the distribution of elements of multiplicative subgroups G of the
multiplicative group Z∗n of the residue ring Zn of n elements, where n is a positive
integer.
Given a multiplicative subgroup G of order t, we consider its coset in Z∗n , or
multiplicative translate, A. For an integer K and a positive integer k, we denote
J(n, A, k, K) = #{ξ ∈ {K + 1, . . . , K + k} : ξ ∈ A}.
Our estimates for J(n, A, k, K) are based on new bounds for the size of the
product sets
AB = {ab : a ∈ A, b ∈ B}
where A, B are sets of rational numbers. The estimates are applied to the following
problems.
1. The structure of substrings in the g-ary expansion of 1/p for almost all primes
p.
2. The asymptotic behaviour in average of the function F (p),the number of
solutions to the congruence
g h ≡ h(modp),
1 ≤ g, h ≤ p − 1.
3. An estimate for the smallest value of positive integers a for which
ap−1 6≡ 1(modp2 ).
The talk is based on joint papers of the speaker with Jean Bourgain, Kevin Ford,
and Igor Shparlinski.
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