Almost primes whose expansion in base \(r\) misses some digits (Q1976808)

This paper is devoted to establishing the existence of almost primes, having at most \(k=k(r)\) prime factors, among the integers whose representation in base~\(r\) does not include certain specified digits. Some of their results apply to numbers whose digits to base~\(r\) are all either 0 or~1. By means of the linear sieve they obtain such a result with \(k(3)=4\), \(k(4)=5\), and where \(k(r) \sim 8r/\pi\) as \(r \to \infty\). If the number \(t\) of permitted digits is somewhat larger then better values for \(k(r)\) follow from using the sieve with weights as described by the reviewer in [Number Theory, Trace Formulas and Discrete Groups, Academic Press, Boston, 289-308 (1989; Zbl 0682.10038)]. If \(t > \sqrt r\) then \(k(r)=5\) is allowed, and if \(t \geq r^{0.68381}\) then \(k(r)=3\) is obtained. The results on the associated ``level of distribution'' needed for these applications of sieve methods are obtained by developing techniques used by \textit{E. Fouvry} and \textit{C. Mauduit} [Acta Arith. 77, 339-351 (1996; Zbl 0869.11073)] and by \textit{P. Erdős, C. Mauduit} and \textit{A. Sárközy} [Discrete Math. 200, 149-154 (1999; Zbl 0945.11006)].