On some results of Korobov and Larcher

Zaremba's famous conjecture (1972) arose from the theory of
numerical integration and relates to the field of continued fractions. It
predicts that for any given prime p there is a positive integer a < p such
that when expanded as a continued fraction $a/p = 1/c_1+1/c_2 +...  + 1/c_s$
all partial quotients $b_j$ are bounded by a constant M. At the moment the
question is widely open although the area has a rich history of works by
Korobov, Hensley, Niederreiter, Bourgain, Kontorovich and many others.
Korobov (1963) proved that one can take $M = O(\log p)$, and in 2022
Moshchevitin--Murphy--Shkredov used the growth in groups and
multiplicative combinatorics to obtain that $M=O(\log p/\log \log p)$.
Applying an additional idea of Dyatlov--Zahl (2016) and Bourgain--Dyatlov
(2018) on the combinatorial structure of Ahlfors--David sets, we show that
the choice $M = O((\log p)^{1/2+o(1)})$ is possible and $O((\log p)^{1/2+o(1)})$
is the limit of the method. Also, we show that there is $a<p$,
$a/p = 1/c_1+1/c_2 +...  + 1/c_s$ such that $s^{-1} \sum_{j=1}^s c_j \ll
\sqrt{\log \log p}$, improving an old result of Larcher (1986).