Claims

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1936_01_01_raikov: Raikov proves that for every integrable f of period 1 and every integer a >= 2 the averages of f(a^k x) converge to the integral of f almost everywhere, so for n_k = a^k no Fourier-tail condition is needed.

2026_04_20_ho: For every C > 0 a mean-zero function in every finite L^p and a lacunary sequence with Fourier tail O((log log log N)^(-C)) whose lacunary averages diverge almost everywhere, so no absolute C exists.

2026_09_26_yang: A {0,1}-valued function and a sequence with n_(j+1) >= 2 n_j whose Fourier tail is at most 8/sqrt(log log m) while its lacunary averages have limit superior at least 3/4 almost everywhere.

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