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Maximal Arithmetic Progressions in Random Subsets
, 2007
"... Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1} N. By an application of the ChenStein method, we show that U (N) −2log N / log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of ar ..."
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Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1} N. By an application of the ChenStein method, we show that U (N) −2log N / log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U (N) /log N converges almost surely to 2 / log 2, while W (N) /log N does not converge almost surely (and in particular, lim sup W (N) /log N ≥ 3/log 2). 1 Introduction and Statement of Results In this note we study the length of maximal arithmetic progressions in a random uniform subset of {0, 1} N. That is, let ξ1, ξ2,..., ξN be a random word in {0, 1} N, chosen uniformly. Consider the (random) set ΞN of elements i such that ξi = 1. Let U (N) denote the maximal length arithmetic progression in ΞN, and let W (N) denote the maximal length aperiodic
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"... Using the Jiang function we find the best theory of arbitrarily long arithmetic progressions of primes 1 Theorem. The fundamental theorem in arithmetic progression of primes. We define the arithmetic progression of primes [13]. P i�1 � P1 ..."
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Using the Jiang function we find the best theory of arbitrarily long arithmetic progressions of primes 1 Theorem. The fundamental theorem in arithmetic progression of primes. We define the arithmetic progression of primes [13]. P i�1 � P1
ELECTRONIC COMMUNICATIONS in PROBABILITY MAXIMAL ARITHMETIC PROGRESSIONS IN RAN
, 2007
"... extreme type limit distribution Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1} N. By an application of the ChenStein method, we show that U (N) − 2log N/log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds fo ..."
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extreme type limit distribution Let U (N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1} N. By an application of the ChenStein method, we show that U (N) − 2log N/log 2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W (N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U (N) /log N converges almost surely to 2/log 2, while W (N) /log N does not converge almost surely (and in particular, limsup W (N) /log N ≥ 3/log 2). 1 Introduction and Statement of Results In this note we study the length of maximal arithmetic progressions in a random uniform subset of {0,1} N. That is, let ξ1,ξ2,...,ξN be a random word in {0,1} N, chosen uniformly. Consider the (random) set ΞN of elements i such that ξi = 1. Let U (N) denote the maximal length arithmetic progression in ΞN, and let W (N) denote the maximal length aperiodic arithmetic