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UID:7166@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20181211T110000
DTEND;TZID=Europe/Paris:20181211T120000
DTSTAMP:20241120T203438Z
URL:https://www.i2m.univ-amu.fr/evenements/zaremba-s-conjecture-and-additi
 ve-combinatorics/
SUMMARY: Ilya Shkredov (Steklov Mathematical Institute\, Moscow): Zaremba's
  conjecture and additive combinatorics
DESCRIPTION: Ilya Shkredov: A well-known Zaremba's conjecture from the theo
 ry of continued fractions says that for any positive integer {q} there is 
 an integer {a}\, 0 &lt\; {a} &lt\; {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 hypothes
 is is open (excepting some particular cases) although in the direction var
 ious results were obtained by Korobov\, Niderreiter\, Bourgain--Kontorovic
 h\, Kan--Frolenkov and others.\nUsing a technique from additive combinator
 ics (we apply results on growth in the group SL2({{F}}p))\, we obtain an e
 xact 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 t
 hat a certain improvement of our upper bound implies the required lower bo
 und.\nhttps://www.youtube.com/watch?v=MQdFczp6tPE
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/Ilya_Shkredov.jpg
CATEGORIES:Séminaire,Ernest
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