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SUMMARY:Manuel Hauke (TU Graz)
DTSTART:20221129T130000Z
DTEND:20221129T140000Z
DTSTAMP:20260423T021334Z
UID:OWNS/102
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/102/">T
 he asymptotic behaviour of Sudler products</a>\nby Manuel Hauke (TU Graz) 
 as part of One World Numeration seminar\n\n\nAbstract\nGiven an irrational
  number $\\alpha$\, we study the asymptotic behaviour of the Sudler produc
 t defined by $P_N(\\alpha) = \\prod_{r=1}^N 2 \\lvert \\sin \\pi r \\alpha
  \\rvert$\, which appears in many different areas of mathematics.\nIn this
  talk\, we explain the connection between the size of $P_N(\\alpha)$ and t
 he Ostrowski expansion of $N$ with respect to $\\alpha$.\nWe show that $\\
 liminf_{N \\to \\infty} P_N(\\alpha) = 0$ and $\\limsup_{N \\to \\infty} P
 _N(\\alpha)/N = \\infty$\, whenever the sequence of partial quotients in t
 he continued fraction expansion of $\\alpha$ exceeds $7$ infinitely often\
 , and show that the value $7$ is optimal.\n\nFor Lebesgue-almost every $\\
 alpha$\, we can prove more: we show that for every non-decreasing function
  $\\psi: (0\,\\infty) \\to (0\,\\infty)$ with $\\sum_{k=1}^{\\infty} \\fra
 c{1}{\\psi(k)} = \\infty$ and\n$\\liminf_{k \\to \\infty} \\psi(k)/(k \\lo
 g k)$ sufficiently large\, the conditions $\\log P_N(\\alpha) \\leq -\\psi
 (\\log N)$\, $\\log P_N(\\alpha) \\geq \\psi(\\log N)$ hold on sets of upp
 er density $1$ respectively $1/2$.\n
LOCATION:https://researchseminars.org/talk/OWNS/102/
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