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SUMMARY:Bartosz Sobolewski (Jagiellonian University\, Krakow\, Poland)
DTSTART:20220527T150000Z
DTEND:20220527T152500Z
DTSTAMP:20260423T011342Z
UID:CANT2022/48
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2022/48/
 ">Monochromatic arithmetic progressions in binary words associated with pa
 ttern sequences</a>\nby Bartosz Sobolewski (Jagiellonian University\, Krak
 ow\, Poland) as part of Combinatorial and additive number theory (CANT 202
 2)\n\n\nAbstract\nLet $e_v(n)$ denote the number of occurrences of a patte
 rn $v$ in the binary expansion of $n \\in \\mathbb{N}$. In the talk we con
 sider monochromatic arithmetic progressions in the class of words $(e_v(n)
  \\bmod{2})_{n \\geq 0}$ over $\\{0\,1\\}$\, which includes the Thue--Mors
 e word $\\mathbf{t}$ ($v=1$) and a variant of the Rudin--Shapiro word $\\m
 athbf{r}$ ($v=11$). So far\, the problem of exhibiting long progressions a
 nd finding an upper bound on their length has mostly been studied for $\\m
 athbf{t}$ and certain generalizations. We show that analogous results hold
  for $\\mathbf{r}$. In particular\, we prove that a monochromatic arithmet
 ic progression of difference $d \\geq 3$ starting at $0$ in $\\mathbf{r}$ 
 has length at most $(d+3)/2$\, with equality infinitely often. We also com
 pute the maximal length of progressions of differences $2^k-1$ and $2^k+1$
 .\nSome weaker results for a general pattern $v$ are provided as well.\n
LOCATION:https://researchseminars.org/talk/CANT2022/48/
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