Thue-Morse at multiples of an integer

Let t=(tn)n⩾0 be the classical Thue-Morse sequence defined by tn=s2(n)(mod2), where s2 is the sum of the bits in the binary representation of n. It is well known that for any integer k⩾1 the frequency of the letter “1” in the subsequence t0,tk,t2k,… is asymptotically 1/2. Here we prove that for any k there is an n⩽k+4 such that tkn=1. Moreover, we show that n can be chosen to have Hamming weight ⩽3. This is best in a twofold sense. First, there are infinitely many k such that tkn=1 implies that n has Hamming weight ⩾3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s2 is replaced by sb for an arbitrary base b⩾2.