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SUMMARY:Roswitha Hofer (JKU Linz)
DTSTART:20230110T130000Z
DTEND:20230110T140000Z
DTSTAMP:20260423T052931Z
UID:OWNS/105
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWNS/105/">E
 xact order of discrepancy of normal numbers</a>\nby Roswitha Hofer (JKU Li
 nz) as part of One World Numeration seminar\n\n\nAbstract\nIn the talk we 
 discuss some previous results on the discrepancy of normal numbers and con
 sider the still open question of Korobov: What is the best possible order 
 of discrepancy $D_N$ in $N$\, a sequence $(\\{b^n\\alpha\\})_{n\\geq 0}$\,
  $b\\geq 2\,\\in\\mathbb{N}$\, can have for some real number $\\alpha$? If
  $\\lim_{N\\to\\infty} D_N=0$ then $\\alpha$ in called normal in base $b$.
  \n\nSo far the best upper bounds for $D_N$ for explicitly known normal nu
 mbers in base $2$ are of the form $ND_N\\ll\\log^2 N$. The first example i
 s due to Levin (1999)\, which was later generalized by Becher and Carton (
 2019). In this talk we discuss the recent result in joint work with Gerhar
 d Larcher that guarantees $ND_N\\gg \\log^2 N$ for Levin's binary normal n
 umber. So EITHER $ND_N\\ll \\log^2N$ is the best possible order for $D_N$ 
 in $N$ of a normal number OR there exist another example of a binary norma
 l number with a better growth of $ND_N$ in $N$. The recent result for Levi
 n's normal number might support the conjecture that $ND_N\\ll \\log^2N$ is
  the best order for $D_N$ in $N$ a normal number can obtain.\n
LOCATION:https://researchseminars.org/talk/OWNS/105/
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