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SUMMARY:Marc Technau (Graz University of Technology)
DTSTART:20200529T110000Z
DTEND:20200529T120000Z
DTSTAMP:20260423T021358Z
UID:NTdL/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTdL/5/">Met
 ric results on summatory arithmetic functions on Beatty sets and beyond</a
 >\nby Marc Technau (Graz University of Technology) as part of Number theor
 y during lockdown\n\n\nAbstract\nThe \\emph{Beatty set} $\\mathcal{B}(\\al
 pha) = \\lbrace\\\, \\lfloor n\\alpha \\rfloor : n\\in\\mathbb{N} \\\,\\rb
 race$ associated to a real number $\\alpha>1$ may be viewed as a generalis
 ed arithmetic progression (consecutive elements differ by either $\\lfloor
  \\alpha \\rfloor$ or $\\lfloor \\alpha \\rfloor+1$) and there are numerou
 s results in the literature on averages of arithmetically interesting func
 tion $f\\colon\\mathbb{N}\\to\\mathbb{C}$ along such Beatty sets. (Here $\
 \lfloor\\xi\\rfloor$ denotes the integer part of a real number $\\xi$.) Fo
 r fixed $\\alpha$\, the quality of such results is usually intricately lin
 ked to Diophantine properties of $\\alpha$. However\, it turns out that th
 e metric theory is much cleaner: in this talk I will discuss recent joint 
 work with A.\\ Zafeiropoulos showing that\n\n\\[ \\Bigl\\lvert \\sum_{\\su
 bstack{ 1\\leq m\\leq x \\\\ m\\in \\mathcal{B}(\\alpha) }} f(m) - \\frac{
 1}{\\alpha} \\sum_{1\\leq m\\leq x} f(m) \\Bigr\\rvert^2 \\ll_{f\,\\alpha\
 ,\\varepsilon} (\\log x) (\\log\\log x)^{3+\\varepsilon} \\sum_{1\\leq m\\
 leq x} \\lvert f(m) \\rvert^2 \\]\n\nholds for almost all $\\alpha>1$ with
  respect to the Lebesgue measure. This significantly improves a beautiful 
 earlier result due to Abercrombie\, Banks\, and Shparlinski. The proof use
 s a recent Fourier-analytic result of Lewko and {Radziwi\\l\\l} based on t
 he classical Carleson--Hunt inequality. Moreover\, it can be shown that th
 e above result is optimal (up to logarithmic factors) in a suitable sense.
  If time permits\, I shall also discuss ongoing work on Piatetski-Shaprio 
 sequences $\\lbrace\\\, \\lfloor n^c \\rfloor : n\\in\\mathbb{N} \\\,\\rbr
 ace$ ($c>1$) of a related spirit.\n
LOCATION:https://researchseminars.org/talk/NTdL/5/
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