Ho 2026 counterexamples lacunary dilates via dyadic spike

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Boon Suan Ho, Counterexamples for lacunary dilates via dyadic spike blocks. arXiv preprint (2026). arXiv:2604.18535. The arXiv record (https://arxiv.org/abs/2604.18535, read 2026-10-02) names the Creative Commons Attribution 4.0 license. The folder's PDF is arXiv:2604.18535v2 [math.CA] (21 April 2026; 27 pages). Read status: claims checked for Theorem 1.1, Definition 1.2 and Corollary 1.3 (p. 3), Corollaries 1.4 and 1.5 and Theorem 1.6 (p. 4) and Theorem 1.7 (p. 5), each read clause by clause on the page images on 2026-10-07, with the short deductions of the three corollaries from Theorem 1.1 (pp. 20--21); the proofs of Theorems 1.1, 1.6 and 1.7 were not read.

The construction is built from dyadic spikes: functions of mean zero and unit L^2 norm that are large and positive on one short dyadic interval and slightly negative elsewhere. Each block F_k is a multiple of a sum of independent copies phi_d(2^a x) of one spike, f is the sum of the blocks, and the exponents of n_j = 2^{m_j} are chosen in stages of trials, so that for the rare x that hit a spike, that spike is counted many times in one short average; a uniform lower bound on every block keeps the other stages from cancelling that gain (pp. 5--6). Theorem 1.1 produces a real mean-zero f in all L^p, p finite, and a dyadic lacunary sequence with ||f - S_N f||2 << (loglog N)^{-1/2} yet limsup N^{-1} sum{j<=N} f(n_j x) = +infinity almost everywhere. Corollaries 1.3, 1.4 and 1.5 are deduced from it (Section 6): Corollary 1.3 states the same divergence with the tail bound omega(N) for any admissible modulus omega (Definition 1.2), a bound implied by the (loglog N)^{-1/2} one (Lemma 6.1; its proof on p. 20 evaluates omega at exp(exp(2A log A)), while (1.3) as printed on p. 3 reads omega(exp(exp(2 log A))), a weaker condition under which the lemma fails, for instance for omega(N) = exp(-(loglog N)/5)), Corollary 1.4 gives the (logloglog N)^{-C} version, and Corollary 1.5 shows the exponent range c > 1/2 in Matsuyama's positive theorem is sharp. Theorem 1.6 gives, for each finite p >= 2, a mean-zero f in L^p whose partial sums exceed N(log N)^{1/p-epsilon} infinitely often almost everywhere, for every epsilon > 0, and Theorem 1.7 is a bounded companion: for each epsilon in (0,1), the indicator of a set E with |E| < epsilon whose lacunary averages have limsup 1 almost everywhere. For problem 996 Corollary 1.4 is the paper's own stated negative answer to the weak Fourier-tail question, and for problem 995 the case p = 2 of Theorem 1.6 shows the partial sums need not be o(N sqrt(loglog N)) almost everywhere.

Source: https://arxiv.org/abs/2604.18535.

Bears on. #995, #996

Results to transcribe.

• Theorem 1.1: There exist a mean-zero f in every finite L^p and a dyadic lacunary sequence with ||f - S_N f||2 << (loglog N)^{-1/2} but limsup N^{-1} sum{j<=N} f(n_j x) = +infinity a.e.
• Corollary 1.3: For any admissible decreasing modulus omega, there is such an f with ||f - S_N f||_2 << omega(N) and divergent lacunary averages a.e.
• Corollary 1.4: Negative answer to Erdos Problem #996: for every C > 0 the Fourier-tail condition (logloglog N)^{-C} fails to force a.e. convergence of lacunary averages.
• Corollary 1.5: For every 0 < c <= 1/2 there are counterexamples with ||f - S_N f||_2 << (loglog N)^{-c}, in particular at the endpoint c = 1/2, so the range c > 1/2 of Matsuyama's theorem is sharp.
• Theorem 1.6: For each 2 <= p < infinity there is f in L^p with limsup (sum_{j<=N} f(n_j x))/(N(log N)^{1/p-epsilon}) = +infinity a.e., for every epsilon > 0; p = 2 answers the example question of Erdos Problem #995 negatively.
• Theorem 1.7: A bounded companion construction: for each 0 < epsilon < 1, a set E with |E| < epsilon and a lacunary sequence with n_{j+1}/n_j >= 2 such that limsup N^{-1} sum_{j<=N} 1_E(n_j x) = 1 a.e., so the bounded mean-zero function 1_E - |E| has lacunary averages that fail to converge a.e.