# 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?