Erdős problem 480 · problems.science

Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that $\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq 5^{-1/2}\approx 0.447?$ A conjecture of Newman.

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FormalConjectures/ErdosProblems/

480.lean

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This was proved by Chung and Graham [ChGr84], who in fact prove that $\inf_n \liminf_{m\to \infty} n \lvert x_{m+n}-x_m\rvert\leq \frac{1}{c}\approx 0.3944$ where $c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.5353705\cdots$ and $F_m$ is the $m$ th Fibonacci number.

FormalConjectures/ErdosProblems/480.leanErdos480.erdos_480.variants.chung_graham3 linesExact file

1have c := 1 + ∑' (k : ℕ+), 1 / ↑(Nat.fib (2 * ↑k));2∀ (x : ℕ → ℝ),3  (∀ (n : ℕ), x n ∈ Set.Icc 0 1) → ⨅ n, Filter.liminf (fun m => ↑↑n * |x (m + ↑n) - x m|) Filter.atTop ≤ 1 / c

SolvedStatement only, no proof

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