Erdős problem 349

For $0 < \alpha \le 1$ and any $t > 0$ , $(t, \alpha)$ is not a good pair: every term $\lfloor t\alpha^n\rfloor$ lies in the finite interval $[0, \lfloor t\rfloor]$ (since $\alpha^n \le 1$ ), so every subset sum is bounded by the constant $\sum_{i \in [0,\lfloor t\rfloor]} i$ , and no large integer can be a subset sum. A partial result on the open Erdős Problem 349, complementing the $2 < \alpha$ and integer-coefficient cases.

Sources

The source's own material, retained exactly.

12 retained statements36c6d0fef71a

FormalConjectures/ErdosProblems/

349.lean

Retained formal statement1 of 12

For $\alpha > 2$ and any $t > 0$ , the sequence $\lfloor t\alpha^n\rfloor$ is not additively complete; equivalently $(t, \alpha)$ is not a "good pair". A partial result on the open Erdős Problem 349: it complements complete_for_alpha_in_Ioo_one_to_goldenRatio.

The proof is recorded via the formal_proof mechanism rather than written inline, as it exceeds the repository's proof-length guideline.

FormalConjectures/ErdosProblems/349.leanErdos349.alpha_gt_two_not_isGoodPair1 lineExact file

1∀ (t α : ℝ), 0 < t → 2 < α → ¬Erdos349.IsGoodPair t α

SolvedProof has a holeformal conjecturesexternal proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.