The level of distribution of a complex valued sequence b indicates how well
b behaves along arithmetic
progressions nd+a, on average over the common difference d.
Larger values of the level of distribution
correspond to increasingly short, and sparse, arithmetic subsequences.
We prove that the Thue-Morse
sequence 01101001... has level of distribution equal to 1, which is essentially best possible.
This concerns
arithmetic subsequences of length N, and having common difference NR,
where R > 0 is arbitrarily large.
This gives one of the first nontrivial examples of a sequence satisfying an analogue of
the Elliott-Halberstam
conjecture from prime number theory. As an application, we show that each of the two symbols 0 and 1
appears with asymptotic frequency 1/2 in the subsequence of the Thue-Morse sequence indexed
by ⌊nc⌋,
where 1 < c < 2.