The Erdos similarity conjecture for geometric progressions

Theorem 1.1: for every $q\in(0,1)$ and $\eta\in(0,1)$ there is a compact $E_{q,\eta}\subseteq[0,1]$ with $m(E_{q,\eta})>1-\eta$ such that for all $x\in\mathbb R$, $s\ne0$, some $x+sq^n\notin E_{q,\eta}$. So no geometric progression is measure universal; the dyadic companion first proved $q=1/2$. The construction is probabilistic (random routing tables on a finite tree). It does not prove the conjecture for other infinite sets, and makes no claim of one set avoiding all ratios at once.

Theorem 1.1: for every $q\in(0,1)$ and $\eta\in(0,1)$ there is a compact $E_{q,\eta}\subseteq[0,1]$ with $m(E_{q,\eta})>1-\eta$ such that for all $x\in\mathbb R$, $s\ne0$, some $x+sq^n\notin E_{q,\eta}$. So no geometric progression is measure universal; the dyadic companion first proved $q=1/2$. The construction is probabilistic (random routing tables on a finite tree). It does not prove the conjecture for other infinite sets, and makes no claim of one set avoiding all ratios at once.