Erdős problem 254 · problems.science

If $A \subseteq \mathbb{N}$ has unbounded dyadic-shell counts and $\sum_{n \in A} \|\theta n\| = \infty$ for every $0 < \theta < 1$ , must $A$ be complete - is every sufficiently large integer a sum of distinct elements of $A$ ?

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

254.lean

Retained formal statement2 of 3

Cassels [Ca60] proved this under the alternative hypotheses $\lim \frac{\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert}{\log\log x}=\infty$ and $\sum_{n\in A} \{ \theta n\}^2=\infty$ for every $\theta\in (0,1)$ .

FormalConjectures/ErdosProblems/254.leanErdos254.erdos_254.variants.cassels5 linesExact file

1∀ (A : Set ℕ),2  (Filter.Tendsto (fun x => (↑(A ∩ Set.Icc 1 (2 * x)).ncard - ↑(A ∩ Set.Icc 1 x).ncard) / Real.log (Real.log ↑x))3        Filter.atTop Filter.atTop ∧4      ∀ (θ : ℝ), 0 < θ → θ < 1 → ¬Summable fun n => distToNearestInt (θ * ↑↑n) ^ 2) →5    ∀ᶠ (m : ℕ) in Filter.atTop, Erdos254.IsSumOfDistinct A m

SolvedStatement only, no proof

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