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SUMMARY:On some results of Korobov and Larcher [MPIM]
DTSTART:20250729T123000Z
DTEND:20250729T133000Z
DTSTAMP:20260721T193400Z
UID:indico-event-579@math-events.uni-bonn.de
DESCRIPTION:Speakers: Ilya Shkredov (Purdue University/MPIM)\n\nNumber the
 ory lunch seminar \nZaremba's famous conjecture (1972) arose from the the
 ory ofnumerical integration and relates to the field of continued fraction
 s. Itpredicts that for any given prime p there is a positive integer a < p
  suchthat 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 mo
 ment thequestion is widely open although the area has a rich history of wo
 rks byKorobov\, Hensley\, Niederreiter\, Bourgain\, Kontorovich and many o
 thers.Korobov (1963) proved that one can take $M = O(\\log p)$\, and in 20
 22Moshchevitin--Murphy--Shkredov used the growth in groups andmultiplicati
 ve combinatorics to obtain that $M=O(\\log p/\\log \\log p)$.Applying an a
 dditional idea of Dyatlov--Zahl (2016) and Bourgain--Dyatlov(2018) on the 
 combinatorial structure of Ahlfors--David sets\, we show thatthe choice $M
  = O((\\log p)^{1/2+o(1)})$ is possible and $O((\\log p)^{1/2+o(1)})$is th
 e 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).\n \n\nhttps://math-
 events.uni-bonn.de/event/579/
LOCATION:MPIM\, Vivatsgasse\,  7 - Lecture Hall (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/579/
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