A Related Problem to Erdős' similarity conjecture

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?

• Your conjecture is correct. Actually $t$ can be 0. – 喻 良 Sep 18 at 0:45
• @喻良 Thanks! Do you have a reference for that? – Thomas Yang Sep 18 at 0:49
• Here is the most recent survey: jstor.org/stable/44153069. Question 1 was answered. I have a positive answer to Question 2 under certain set theoretical assumptions. – 喻 良 Sep 18 at 0:55
• FYI LaTeX-style accenting like \H{o} doesn't work in text on this site, only in math. If you want ő you have to actually type it (if you don't know how to type it on your keyboard, your OS may have a character map tool). – Nate Eldredge Sep 18 at 1:21
• For $t=0$ almost any $\delta\in (0,1)$ works: not appropriate values are covered by countably many null sets. Well, I guess this may be what 喻 良 essentially says, but without special terminology which I am sorry to be not familiar with. – Fedor Petrov Sep 18 at 7:47

Proof: It is simple to see that if $$\delta$$ is $$x$$-random, then so is $$\delta2^{-n}$$ for any $$n\geq 1$$. But $$E$$ is conull and so there must be a real $$x$$ so that every $$x$$-random belongs to $$E$$. Then it is clear that every $$x$$-random real $$\delta$$ meets your requirement. QED
Actually this method can be used to prove that for any countable set $$A\subset (0,1)$$ , there is a real $$\delta$$ so that $$\delta y\in E$$ for any $$y\in A$$ (The result was proved by Kolountzakis first).