Dyadic spike blocks yield counterexamples where lacunary averages diverge to infinity a.e. for mean-zero functions in all L^q and for L^p functions, resolving Erdős problems #995 and #996 negatively.
62, Société Mathématique de France, Paris, 2024, pp
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For lacunary sequences with a_{n+1} ≥ q a_n, liminf NG_N(x)/log N ≥ 1/2 and limsup ≤ (q+1)/(q-1) a.e.; the limit equals 1 a.e. if a_n divides a_{n+1}.
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Counterexamples for lacunary dilates via dyadic spike blocks
Dyadic spike blocks yield counterexamples where lacunary averages diverge to infinity a.e. for mean-zero functions in all L^q and for L^p functions, resolving Erdős problems #995 and #996 negatively.
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Maximal Gaps for Dilated Lacunary Integer Sequences
For lacunary sequences with a_{n+1} ≥ q a_n, liminf NG_N(x)/log N ≥ 1/2 and limsup ≤ (q+1)/(q-1) a.e.; the limit equals 1 a.e. if a_n divides a_{n+1}.