A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-Term Arithmetic Progressions
نویسنده
چکیده
Assuming the well known conjecture that for any γ > 0 and x sufficiently large the interval [x, x+xγ ] always contains a prime number, we prove the following unexpected result: There exist numbers 0 < ρ < 1 arbitrarily close to 0, and arbitrarily large primes q, such that if S is any subset of Z/qZ of density at least ρ, having the least number of 3-term arithmetic progressions among all such sets S (of density ≥ ρ), then there exists an integer 1 ≤ b ≤ q − 1 and a real number 0 < d < 1 (depending only on ρ) such that
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