The geometric case of the Erdős similarity conjecture

We prove the geometric-progression case of the Erdős similarity conjecture. For every fixed $q\in (0,1)$ and every $\eta \in (0,1)$, we construct a compact set $E_{q,\eta }\subseteq [0,1]$ of measure greater than $1-\eta$ containing no nontrivial affine copy of $\{q^n:n\ge 1\}$, for any translation and either sign of nonzero dilation. The set may depend on q; the result makes no simultaneous assertion for different ratios.