Linear equations in integers with bounded sum of digits (Q922583)

\textit{H. G. Senge} and \textit{E. G. Straus} [Period. Math. Hung. 3, 93-100 (1973; Zbl 0248.12004)] proved that the number of integers, the sum of whose digits in each of the bases a and b are bounded by a fixed c is finite if and only if a and b are multiplicatively independent. An explicit estimate for the number of such integers was obtained by \textit{C. L. Stewart} [J. Reine Angew. Math. 319, 63-72 (1980; Zbl 0426.10008)], using Baker's method. By using his p-adic generalization [J. Reine Angew. Math. 406, 44-108 (1990; Zbl 0693.10027)] of Schmidt's subspace theorem recently the author gave an extension of the finiteness result of Senge and Straus to the case of several variables. He proved that if \(b_ 1,...,b_ k\) are pairwise multiplicatively independent natural numbers, then the number of solutions of \[ (*)\;\pm n_ 1\pm n_ 2\pm...\pm n_ k=0 \] in nonnegative integers \(n_ 1,...,n_ k\), such that the sum of digits of \(n_ i\) in the base \(b_ i\) is bounded by \(c(i=1,...,k)\), is finite. In the present paper, applying his quantitative version of the theorem on S-unit equations [J. Reine Angew. Math. 406, 109-120 (1990; Zbl 0693.10016)] the author gives an explicit bound for the number of such solutions of (*). The bound depends only on the number of prime factors of \(b_ 1,...,b_ k\), on k and on c.