The starting point of this paper is Graham’s problem on the \(p\)-divisibility of central binomial coefficients: are there infinitely many integers \(n\ge1\) such that the binomial coefficient \(\binom{2n}n\) is coprime with \(105\)?
By a result of Kummer, this is equivalent to a question on the simultaneous expansion of \(n\) in bases \(3\), \(5\), and \(7\). Namely, \(\binom{2n}n\) is coprime with \(105\) if and only if the base-\(p\) expansion of \(n\) contains only digits \(\leq (p-1)/2\), for \(p\in\{3,5,7\}\).
The author proves a number of theorems on integers with restricted digits. For example, Theorem 2.5 states there exist infinitely many identities \[ \sum_{j\in I_3}3^j+ \sum_{j\in I_4}4^j= \sum_{j\in I_5}5^j, \] where \(I_3\), \(I_4\), and \(I_5\) are finite subsets of \(\mathbb N\).
Theorem 2.7 states that for any integer \(k\ge1\) there exists \(M\) having the following property. If \(b_1,\ldots,b_k\) are multiplicatively independent integers bounded below by \(M\), there are infinitely many integers \(n\) whose base-\(b_j\) expansion omits the digit zero, for all \(j\in\{1,\ldots,k\}\).
In the proofs, connections to geometric measure theory are exploited, such as Newhouse’s gap lemma.
A conditional result (Theorem 2.2) is also proved, which assumes the validity of Schanuel’s conjecture on the transcendence degree of field extensions of \(\mathbb Q\).