Problem packetResearch packetR698
Determine the growth order of the greedy cube-difference-free set
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Attempt outcome: open strategy
Recorded scope: No scope is recorded.
Originating problem: Density of losing positions in the subtract-a-cube game
Authored record and scope
- Authored title
- Determine the growth order of the greedy cube-difference-free set
- Record type
- attempt
- Stored status
- open_strategy
- Evidence grade
- proposed
Work and source credit
- Recorded action
No action description supplied.
- Authored result summary
Density zero leaves a quantitative problem: estimate the number of subtract-a-cube P-positions through \(N\).
- Reported outcome
No separate outcome supplied.
- Recorded status
open_strategy
- Recorded evidence grade
proposed
- Recorded scope
No explicit scope supplied.
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2Authored explanation
Maximality gives an elementary lower bound. Every heap up to \(N\) is either a P-position or a P-position plus one of the \(\lfloor N^{1/3}\rfloor\) positive cubes, so \(L(N)+1\geq (N+1)/(\lfloor N^{1/3}\rfloor+1)\). Thus \(L(N)=\Omega(N^{2/3})\). The polynomial difference theorem gives \(L(N)=o(N)\). Sharpening that gap for this particular greedy set is the natural successor question. Its literature status still needs a specialist search.
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3Outcome
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Verification source: doi.org ↗, Quantitative direction informed by Balog, Pelikán, Pintz, and Szemerédi, Difference sets without k-th powers, 1994
4How it connects
Uses
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- artifact
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}6Provenance
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