Partial quotients and equidistribution (Q639576)

For a real number \(x\in [0,1]\) let \(x=[a_1,a_2,\dots]\) be its continued fraction. Write \(a_i(x):=a_i\) to emphasize the dependence on \(x\). A conjecture of \textit{S. K. Zaremba} [Appl. Number Theory numer. Analysis, Proc. Sympos. Univ. Montreal 1971, 39--119 (1972; Zbl 0246.65009)] asserts that there exists a constant \(c>0\) such that for all positive integers \(N\) there exists \(b\in \{1,\dots,N\}\) coprime to \(N\) such that \(\max\{a_i(b/N):i=1,2,\dots\}\leq c\). If true, this would imply that \[ \sum_{i} a_i(b/N)\leq c\log N \] for such a number \(b\). The best partial result known is a result of \textit{G. Larcher} [Monatsh. Math. 101, 135--150 (1986; Zbl 0584.10030)] to the effect that there exists a \(b\in \{1,\dots,N\}\) coprime to \(N\) such that the above inequality holds with an extra \(\log\log N\) factor in the upper bound. In the paper under review, the author shows that Larcher's result holds when \(N=p\) is prime not only for one \(b\) but for most primitive \(b\) roots modulo \(p\). In fact, it even holds for almost all elements of any multiplicative subgroup \(G\) of \({\mathbb Z}_p^*\) of cardinality at least \(p^{7/8+\varepsilon}\), where \(\varepsilon>0\) is arbitrary. The result has implications for the statistical properties of the pseudo-congruential number generator modulo \(p\).