On the distribution of the roots of certain congruences and a problem for additive arithmetic functions

The asymptotic distribution of the roots of the congruence ax ≡ b (mod D), 1 ≤ x ≤ D, as D varies, is investigated. Quantitative estimates are obtained by means of exponential sums combined with sieve methods. As an application of the results it is shown that if an additive arithmetic function satisfies f(an + b) ? f(cn + d) = O(1) for all positive integers n, ad ≠ bc, then f(n) = O((log n)³) must hold. This result is apparently the first bound of any kind in such a situation.