Erd\H os' similarity conjecture states that for each infinite set $A\subset \mathbb R$ there is a set $P\subset [0,1]$ of positive measure such that for all $t\in \mathbb R$, $\delta\neq 0$ there is some $a\in A$ with $t+\delta a\notin P$. In particular, it is unknown if the sequence $A=\{2^{-n}:n\geq 1\}$ has the property above.

I am considering the following related proposition (not sure if this has been answered in previous literature):

If $E\subset [0,1]$ has full measure then there is $t\in \mathbb R$ and $\delta\neq 0$ such that $t+\delta 2^{-n}\in E$ for all $n\geq 1$.

Is this correct?

Erdős' similarity conjecture states that for each infinite set $A\subset \mathbb R$ there is a set $P\subset [0,1]$ of positive measure such that for all $t\in \mathbb R$, $\delta\neq 0$ there is some $a\in A$ with $t+\delta a\notin P$. In particular, it is unknown if the sequence $A=\{2^{-n}:n\geq 1\}$ has the property above.

I am considering the following related proposition (not sure if this has been answered in previous literature):

If $E\subset [0,1]$ has full measure then there is $t\in \mathbb R$ and $\delta\neq 0$ such that $t+\delta 2^{-n}\in E$ for all $n\geq 1$.

Is this correct?