Sums of the Thue-Morse sequence over arithmetic progressions
Cusick, Thomas W.
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In this noet we use the theory of Boolean functions to find a new elementary proof for Moser's conjecture that states that in the bounded sequence of nonnegative integers divisible by 3 there are more integers with an even number of Is in their base-2 representation. This proof is simpler than the original proof by D. J. Neuman in 1969. We further apply the method to prove a similar result for p = 5, which was also done by Grabner in 1993. The methods seem to be extendable to other primes, but the computations for the relevant constants will be quite complex.
Advances and Applications in Discrete Mathematics 4:2 (2009), 127-135.