The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit
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چکیده
Given an integer q ≥ 2 and a number θ ∈ (0, 1], consider the collection of all subsets of Zq := Z/qZ having at least θq elements. Among the sets in this collection, suppose S is any one having the minimal number of three-term arithmetic progressions, where in our terminology a three-term arithmetic progression is a triple (x, y, z) ∈ S3 satisfying x + y ≡ 2z (mod q). Note that this includes trivial progressions (x, x, x), and note that the progression (x, x+ d, x+ 2d) is different from (x+ 2d, x+ d, x). Now, let
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The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit
How few three-term arithmetic progressions can a subset S ⊆ ZN := Z/NZ have if |S| ≥ υN? (that is, S has density at least υ). Varnavides [4] showed that this number of arithmetic-progressions is at least c(υ)N for sufficiently large integers N ; and, it is well-known that determining good lower bounds for c(υ) > 0 is at the same level of depth as Erdös’s famous conjecture about whether a subset...
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