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Erdős problem 254

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If A⊆NA \subseteq \mathbb{N} has unbounded dyadic-shell counts and ∑n∈A∥θn∥=∞\sum_{n \in A} \|\theta n\| = \infty for every 0<θ<10 < \theta < 1, must AA be complete - is every sufficiently large integer a sum of distinct elements of AA?

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4 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

254.lean

Retained formal statement2 of 3

Cassels [Ca60] proved this under the alternative hypotheses lim⁡∣A∩[1,2x]∣−∣A∩[1,x]∣log⁡log⁡x=∞\lim \frac{\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert}{\log\log x}=\infty and ∑n∈A{θn}2=∞\sum_{n\in A} \{ \theta n\}^2=\infty for every θ∈(0,1)\theta\in (0,1).

FormalConjectures/ErdosProblems/254.leanErdos254.erdos_254.variants.cassels5 linesExact file
∀ (A : Set ℕ),  (Filter.Tendsto (fun x => (↑(A ∩ Set.Icc 1 (2 * x)).ncard - ↑(A ∩ Set.Icc 1 x).ncard) / Real.log (Real.log ↑x))        Filter.atTop Filter.atTop ∧      ∀ (θ : ℝ), 0 < θ → θ < 1 → ¬Summable fun n => distToNearestInt (θ * ↑↑n) ^ 2) →    ∀ᶠ (m : ℕ) in Filter.atTop, Erdos254.IsSumOfDistinct A m
SolvedStatement only, no proof