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SUMMARY:On the number of arithmetic progressions in sparse random sets - C
 hristoph Koch (Department of Statistics\, University of Oxford)
DTSTART;VALUE=DATE-TIME:20180423T120000
DTEND;VALUE=DATE-TIME:20180423T130000
UID:https://talks.ox.ac.uk/talks/id/50cd5e62-d126-4531-a276-f7ed5417f890/
DESCRIPTION:We study arithmetic progressions $\\{a\,a+b\,a+2b\,\\dots\,a+(
 \\ell-1) b\\}$\, with $\\ell\\ge 3$\, in random subsets of the initial seg
 ment of natural numbers $[n]:=\\{1\,2\,\\dots\, n\\}$. Given $p\\in[0\,1]$
  we denote by $[n]_p$ the random subset of $[n]$ which includes every numb
 er with probability $p$\, independently of one another. The focus lies on 
 sparse random subsets\, i.e.\\ when $p=p(n)=o(1)$ with respect to $n\\to\\
 infty$.\n\nLet $X_\\ell$ denote the number of distinct arithmetic progress
 ions of length $\\ell$ which are contained in $[n]_p$. We determine the li
 miting distribution for $X_\\ell$ not only for fixed $\\ell\\ge 3$ but als
 o when  $\\ell=\\ell(n)\\to\\infty$ sufficiently slowly. Moreover\, we pro
 ve a central limit theorem for the joint distribution of the pair $(X_{\\e
 ll}\,X_{\\ell'})$ for a wide range of $p$. Our proofs are based on the met
 hod of moments and combinatorial arguments\, such as an algorithmic enumer
 ation of collections of arithmetic progressions.\n\nThis is joint work wit
 h Yacine Barhoumi-Andr\\'eani and Hong Liu (University of Warwick).\nSpeak
 ers:\nChristoph Koch (Department of Statistics\, University of Oxford)
LOCATION:Mathematical Institute (L4)\, Woodstock Road OX2 6GG
TZID:Europe/London
URL:https://talks.ox.ac.uk/talks/id/50cd5e62-d126-4531-a276-f7ed5417f890/
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DESCRIPTION:Talk:On the number of arithmetic progressions in sparse random
  sets - Christoph Koch (Department of Statistics\, University of Oxford)
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