co.combinatorics – Well approximating sets

Given a real number $x in (0, 1)$, we denote by $0.x_1x_2…$ its binary expansion, where we always choose the expansion that ends in an infinite number of $1’s$ whenever a choice has to be made.

Given two real numbers $a = 0.a_1a_2…$ and $b = 0.b_1b_2…$ in $(0, 1)$, denote by $S(a, b)$ the set of indices $i$ for which $a_i neq b_i$. We label the elements of $S(a, b)$ by $s_0, s_1, …$ where the values of the sequence are chosen to be in increasing order. We suppress the dependence on $a, b$ when writing the sequence for clarity.

We say an increasing sequence $c_n$ of positive real numbers is anti-syndetic if for any polynomial $P$ with positive coefficients, there exists some $N in mathbb N$ such that $c_{n + N} > P(n)$ for all $n in mathbb N$.

A subset $D$ of $(0, 1)$ is said to be well approximating if for every $x in (0, 1)$, there is a $d in D$ such that the sequence $s_n$ associated with $S(x, d)$ is anti-syndetic.

Question: Does there exist a well approximating subset of $(0, 1)$ with Lebesgue measure zero?