The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit

Given a density t in (0,1], and a prime p, let S be any subset of F_p having at least tp elements, and having the least number of three-term arithmetic progressions mod p among all subsets of F_p with at least tp elements. Define N(t,p) to be 1/p^2 times the number of three-term arithmetic progressions in S modulo p. Note that N(t,p) does not depend on S -- it only depends on t and p. An old result of Varnavides shows that for fixed t, N(t,p) > c(t) > 0 for all primes p sufficiently large. But, does N(t,p) converge to a limit as p -> infinity? We prove that it does.