The distribution \(\bmod n\) of fractions with unbounded partial quotients (Q1891177)

This paper concerns (reduced) fractions with bounded partial quotients, and the distribution \(\text{mod } n\) of the corresponding integer pairs (numerator, denominator)=\((a,b)\) say. With or without a bound on the partial quotients, there are certain pairs of congruence classes \(\text{mod } n\) that are forbidden to such pairs, e.g. \((0,0)\). There are \(n^ 2 \prod_{p\mid n} (1-p^{-2})\) possible values for \((a,b)\bmod n\). If there is no constraint on the partial quotients, an elementary inclusion and exclusion argument shows that the asymptotic distribution, taken over all fractions with \(1\leq a<b\leq x\), of \((a,b)\bmod n\) is uniform across all available congruence class pairs. The main result is that, for any \(k>1\), and when attention is restricted to fractions for which all partial quotients are \(\leq k\), this asymptotically uniform distribution persists. The proof requires recently established asymptotic estimates for the number of such fractions with \(1\leq a<b\leq x\) (as a function also of \(k\)), and then a `mixing' argument that takes the place of inclusion and exclusion. The author seems to think (as does the reviewer, who is the author) that this result lends support, with \(k=2\), to Zaremba's conjecture to the effect that there exists \(k\) such that for all sufficiently large \(b\), there exists an \(a\) so that all partial quotients in the continued fraction expansion of \(a/b\) are \(\leq k\).