Remark: Family 084. The geometric case of the Erd (o)s similarity conjecture

For every fixed $q\in (0,1)$, constructs compact subsets of $[0,1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $\{q^n:n\ge 1\}$, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.