Zaremba’s conjecture and additive combinatorics

Ilya Shkredov
Steklov Mathematical Institute, Moscow
http://www.mi-ras.ru/~ishkredov/

Date(s) : 11/12/2018   iCal
11 h 00 min - 12 h 00 min

A well-known Zaremba’s conjecture from the theory of continued fractions says that for any positive integer {q} there is an integer {a}, 0 < {a} < {q}, which is relatively prime to {q} such that for the finite continued fraction expansion of the rational {a}/{q} = [{x}1, …, {x}s] one has {x}{j} ≤ 5. At the moment the hypothesis is open (excepting some particular cases) although in the direction various results were obtained by Korobov, Niderreiter, Bourgain–Kontorovich, Kan–Frolenkov and others.
Using a technique from additive combinatorics (we apply results on growth in the group SL2({{F}}p)), we obtain an exact upper bound for cardinality of Zaremba’s numbers {a}, i.e. such {a}∈{{F}}p) for which Zaremba’s conjecture takes place. Besides, we show that a certain improvement of our upper bound implies the required lower bound.

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