Linear Congruences in Continued Fractions on Finite Alphabets

Abstract

A linear homogeneous congruence ay ≡ bY (mod q) is considered and an order-sharp upper bound for the number of its solutions is proved. Here a, b, and q are given jointly coprime numbers and y and Y are coprime variables in a given closed interval such that the number y/Y can be expanded in a continued fraction with partial quotients from some alphabet A ⊆ ℕ. For A = ℕ (and without the assumption that y and Y are coprime), a similar problem was solved by N. M. Korobov.