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Math-2947 · mathematics · Analysis

Avoiding every geometric affine sequence in a large-measure set

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $q\in(0,1)$ and every $\eta\in(0,1)$, is there a compact $E\subset[0,1]$ of Lebesgue measure greater than $1-\eta$ such that no set \[ \{a+bq^n:n\geq1\}, \] with $a$ real and $b$ a nonzero real of either sign, is contained in $E$?

Short English statement

For every $q\in(0,1)$ and $\eta\in(0,1)$, does some compact $E\subset[0,1]$ of Lebesgue measure greater than $1-\eta$ avoid containing $\{a+bq^n:n\geq1\}$ for every real $a$ and every nonzero real $b$ of either sign?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2947-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2947

For each fixed $q\in(0,1)$ and each positive measure tolerance, the theorem constructs a compact subset of $[0,1]$ avoiding every affine copy of $\{q^n:n\ge1\}$ with nonzero dilation of either sign. Its measure exceeds one minus the tolerance. This is exactly the geometric-progression assertion recorded here, with the avoiding set allowed to depend on both parameters.

Avoiding every geometric affine sequence in a large-measure set

OpenAI's claim

Avoiding a fixed geometric progression: For each fixed ratio $q$ strictly between zero and one, the manuscript claims almost-full-measure compact sets in the unit interval avoiding every affine copy of $(q^n)$. The chosen set is permitted to depend on the ratio.

Mathematical statement of the claimed result

Mathematical claim 1.
For any fixed $0<q<1$ and $0<\eta<1$ the claimed construction produces compact $E_{q,\eta}\subset[0,1]$ with $m(E_{q,\eta})>1-\eta$ and \[\forall x\in\mathbb R\ \forall s\in\mathbb R\setminus\{0\}\ \exists n\ge1:\quad x+sq^n\notin E_{q,\eta}.\] The avoiding set may depend on both $q$ and $\eta$.

Outline of the argument

Avoiding a fixed geometric progression: The construction adapts the ordered-tree random hitting method, using nested grids with resolutions matched to the chosen geometric ratio. Gaps between edge index intervals preserve previous routing decisions and expose independent current trials. The number of scale representatives is controlled locally within each subtree, and a final neighborhood repair deals with centers missed by the random construction.

Formal verification

Family 084 describes Lean coverage for avoidance for the dyadic ratio; it does not assert avoidance for an arbitrary geometric ratio.

Avoiding a fixed geometric progression: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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