ON THE MAXIMAL NUMBER OF THREE-TERM ARITHMETIC PROGRESSIONS IN SUBSETS OF Z/pZ
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چکیده
Let α ∈ [0, 1] be a real number. Ernie Croot [3] showed that the quantity max A⊆Z/pZ |A|=⌊αp⌋ #(3-term arithmetic progressions in A) p tends to a limit as p → ∞ though primes. Writing c(α) for this limit, we show that c(α) = α/2 provided that α is smaller than some absolute constant. In fact we prove rather more, establishing a structure theorem for sets having the maximal number of 3-term progressions amongst all subsets of Z/pZ of cardinality m, provided that m < cp.
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تاریخ انتشار 2008