Erdős problem 996 · problems.science

Let $n_1<n_2<\cdots$ be a lacunary sequence of integers and $f\in L^2([0,1])$ with $n$ th Fourier partial sum $f_n$ . Is there an absolute constant $C>0$ such that if $\| f-f_n\|_2 \ll (\log\log\log n)^{-C}$ then $\frac{1}{N}\sum_{k\leq N}f(\{\alpha n_k\})\to\int_0^1 f$ for almost every $\alpha$ ? A preprint answers this negatively via a dyadic spike-block counterexample.

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FormalConjectures/ErdosProblems/

996.lean

Retained formal statement2 of 2

The following theorem is proved in [Ma66].

FormalConjectures/ErdosProblems/996.leanErdos996.erdos_996.variants.log210 linesExact file

1∀ (C : ℝ),2  0.5 < C →3    ∀ (f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)) (n : ℕ → ℕ),4      IsLacunary n →5        ((fun k =>6              (MeasureTheory.eLpNorm (Erdos996.fourierPartial f k) 2 AddCircle.haarAddCircle).toReal) =O[Filter.atTop]7            fun k => 1 / Real.log (Real.log ↑k) ^ C) →8          ∀ᵐ (x : AddCircle 1),9            Filter.Tendsto (fun N => (∑ k ∈ Finset.range N, ↑↑f (n k • x)) / ↑N) Filter.atTop10              (nhds (∫ (t : AddCircle 1), ↑↑f t ∂AddCircle.haarAddCircle))

SolvedStatement only, no proof

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