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SUMMARY:Oleh Makarchuk (Institute of Mathematics\, Natl. Acad. Sci. Ukrain
 e)
DTSTART:20240208T133000Z
DTEND:20240208T150000Z
DTSTAMP:20260423T004733Z
UID:fran/48
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/fran/48/">On
  some metric and probabilistic problems given in terms of the $A_2$-contin
 ued fraction representation</a>\nby Oleh Makarchuk (Institute of Mathemati
 cs\, Natl. Acad. Sci. Ukraine) as part of Семінар з фрактал
 ьного аналізу / Fractal analysis seminar\n\n\nAbstract\nIn the
  talk\, we consider the $A_2$-continued fraction expansion with alphabet $
 \\{\\alpha_1\, \\alpha_2\\}$\, where $\\alpha_2 > \\alpha_1 > 0$\, $\\alph
 a_2 \\alpha_1 = 0.5$. Let $\\nu(\\alpha_2\, n\, x)$ be a relative frequenc
 y of digit $\\alpha_2$ among the first $n$ digits of the $A_2$-continued f
 raction representation of a number $x = [a_1\, a_2\, \\ldots\, a_n\, \\ldo
 ts]$. We show that\, for $\\alpha_2 > 8 - 4 \\sqrt{2}$ and for some $C \\i
 n (0\, 1)$\, condition $\\limsup_{n \\to +\\infty} \\nu(\\alpha_2\, n\, x)
  \\le C$ holds for almost all (with respect to Lebesgue measure) numbers $
 x \\in [\\alpha_1\, \\alpha_2]$.\n\nLet $(\\Omega\, S\, P(\\cdot))$ be a p
 robability space and let $(\\xi_n)$ be a given sequence of independent dis
 crete random variables taking values $\\alpha_1$ and $\\alpha_2$ with prob
 abilities $0.5$\, respectively. For random variable $\\xi = [\\xi_1\, \\xi
 _2\, \\ldots\, \\xi_n\, \\ldots]$\, we consider conditions for denominator
 s $q_n(\\xi(\\omega))$ of convergents of $\\xi$ that hold for almost all $
 \\omega \\in \\Omega$.\n
LOCATION:https://researchseminars.org/talk/fran/48/
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