Sums of a smooth number and a number with missing digits (Q2874925)

The subject of this paper is the number of representations of a natural number \(n\) as \(n=s+d\), where \(s\) is a smooth number and \(d\) is a number with missing digits, roughly speaking, and the main result is an asymptotic formula for the number of such representations valid for almost all \(n\). The precise statement of the theorem is as follows.NEWLINENEWLINELet \(g\) be a natural number exceeding 3, and \(u\) be a real number \(\geq1\). Then write \({\mathcal S}(m,u)\) for the set of natural numbers \(s\) up to \(g^m\) (\(=N\), say) such that the largest prime factor of \(s\) is at most \(N^{1/u}\). Next let \(D\) be a subset of \(\{0,1,2,\cdots,g-1\}\) such that \(|D|\geq3\), \(0\in D\), and no prime number divides all members of \(D\). Then let \(W_D(t)\) be the set of natural numbers \(d\) up to \(t\) such that the \(g\)-ary expansion of \(d\) takes the shape \(d=\sum_j c_jg^j\) with \(c_j\in D\). Now write \(R(n)\) for the number of pairs \((s,d)\) satisfying \(n=s+d\) with \(s\in{\mathcal S}(m,u)\) and \(d\in W_D(n)\). Then the main theorem of the paper asserts that there exist functions \(\delta_0=\delta_0(m,u)\) and \(\varepsilon_0=\varepsilon_0(m,u)\), both tending to 0 as \(m\rightarrow\infty\), such that the following formula holds for every \(n\leq N\) with at most \(\varepsilon_0N\) exceptions: NEWLINE\[NEWLINE R(n)=|W_D(N)|\rho(u)\bigl(1+O(\delta_0)\bigr), NEWLINE\]NEWLINE where \(\rho(u)\) denotes the Dickman function. The proof is based on the circle method.NEWLINENEWLINEFor the entire collection see [Zbl 1279.00053].