The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit
نویسنده
چکیده
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 T of the naturals where ∑ n∈T 1/n diverges, has a k-term arithmetic progression for k = 3 (that is, a three-term arithmetic progression). The author answers a question of B. Green [1] about how this minimial number of progressions oscillates for a fixed density υ as N runs through the primes, and as N runs through the odd positive integers.
منابع مشابه
The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit
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 tri...
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