Arithmetically random bitstreams

The Thue–Morse sequence is such an example, as was (first, I believe) proved by Dumont.

If you take a uniformly random real number in $[0,1]$, its binary expansion will have this property with probability $1$; I imagine it is conjectured that the binary expansion of every algebraic irrational number has this property.

You might also be interested in the related Erdös discrepancy problem.

The Champernowne constant $C_2$ ( see https://en.wikipedia.org/wiki/Champernowne_constant ) has the stronger property of normality (see https://en.wikipedia.org/wiki/Normal_number#Properties) for properties of normal numbers. If you examine a normal number along an infinite arithmetic progression and extract the resulting digits, this is also a normal number

Mauduit and Sarkozy have studied essentially this and other related pseudorandomness measures for finite as well as infinite $\{\pm 1\}-$valued sequences, see here (not paywalled)and the references therein.

Briefly, for a finite sequence $(e_1,\ldots, e_N)\in \{\pm 1\}^N$ of length $N,$ they define the well-distribution measure of the sequence by $$ W(e_1,\ldots,e_N)=\max_{a,b,t \in \mathbb{N}} \left| \sum_{j=0}^{t-1} e_{a+jb} \right| $$ where the maximum is taken over all AP's within $\{1,2,\ldots,N\}$.

Another measure they define is the correlation measure of order $k$

$$ C_k(e_1,\ldots,e_N)=\max_{M,0\leq d_1<d_2<\ldots<d_k} \left| \sum_{n=1}^M e_{n+d_1} e_{n+d_2} \cdots e_{n+d_k} \right| $$ with $M+d_k\leq N.$ They prove that for any sequence,

$$ W(e_1,\ldots,e_N) \leq \sqrt{3 N C_2(e_1,\ldots,e_N)} $$ while for almost all sequences in $\{\pm 1\}^N$ one has $$ \sqrt{N} \ll C_2(e_1,\ldots,e_N) \ll \sqrt{N\log N} $$

They also consider Champerpowne, Thue-Morse, and other sequences, with respect to these measures.