Products of integers with few nonzero digits

Abstract

Let s(n) be the number of nonzero bits in the binary digital expansion of the integer n. We study, for fixed k, ℓ, m, the Diophantine systemand s(b) = m, in odd integer variables a, b. When k = 2 or k = 3, we establish a bound on ab in terms of ℓ and m. While such a bound does not exist in the case of k = 4, we give an upper bound for min{a, b} in terms of ℓ and m.