The Erdos similarity conjecture for geometric progressions

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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. parent of this event

VibeMathed record: The Erdos similarity conjecture for geometric progressions evidence for this event

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Companion: The dyadic case of the Erdos similarity conjecture evidence for this event

The Erdos similarity conjecture for geometric progressions evidence for this event

The Erdos similarity conjecture for geometric progressions parent of this event

Lean proof of the dyadic case only (OAI.Problem310.dyadic_affine_avoidance) evidence for this event