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Семинар Добрушинской лаборатории Высшей школы современной математики МФТИ
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Zaremba's conjecture and growth in groups И. Д. Шкредов Институт проблем передачи информации им. А.А. Харкевича Российской академии наук, г. Москва |
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Аннотация: Zaremba's conjecture belongs to the area of continued fractions. It predicts that for any given positive integer q there is a positive a, a<q, (a,q)=1 such that all partial quotients b_j in its continued fractions expansion a/q = 1/b_1+1/b_2 +... + 1/b_s are bounded by five. At the moment the question is widely open although the area has a rich history of works by Korobov, Hensley, Niederreiter, Bourgain and many others. We survey certain results concerning this hypothesis and show how growth in groups helps to solve different relaxations of Zaremba's conjecture. In particular, we show that a deeper hypothesis of Hensley concerning some Cantor-type set with the Hausdorff dimension >1/2 takes place f or the so-called modular form of Zaremba's conjecture. Ссылка для подключения: https://zoom.us/j/93175142429?pwd=VDViRHNOSlZSVUM5ZU03SGZyZy8xQT09 Id: 931-7514-2429 passw=057376 |