Abstract. We prove that the Thue-Morse sequence t along subsequences indexed by ⌊n c ⌋ is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1} L , where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence n → t ⌊n c ⌋ has asymptotic density 2 −L . This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3.In particular, this result shows that for 1 < c < 3/2 the sequence n → t ⌊n c ⌋ attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author.In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue-Morse sequence, that is, it satisfies a Bombieri-Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1} L is contained as an arithmetic subsequence of t.

Abstract. For a nonnegative integer t, let ct be the asymptotic density of natural numbers n for which s(n + t) ≥ s(n), where s(n) denotes the sum of digits of n in base 2. We prove that ct > 1/2 for t in a set of asymptotic density 1, thus giving a partial solution to a conjecture of T. W. Cusick stating that ct > 1/2 for all t. Interestingly, this problem has several equivalent formulations, for example that the polynomial X(X + 1) · · · (X + t − 1) has less than 2 t zeros modulo 2 t+1 . The proof of the main result is based on Chebyshev's inequality and the asymptotic analysis of a trivariate rational function using methods from analytic combinatorics.

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The level of distribution of a complex-valued sequence $b$ measures the quality of distribution of $b$ along sparse arithmetic progressions $nd+a$ . We prove that the Thue–Morse sequence has level of distribution $1$ , which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri–Vinogradov-type theorem for each exponent $\theta <1$ . This result improves on the level of distribution $2/3$ obtained by Müllner and the author. As an application of our method, we show that the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$ , where $1 < c < 2$ , is simply normal. This result improves on the range $1 < c < 3/2$ obtained by Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue–Morse sequence along the squares is simply normal.

Abstract. Integer sequences of the form ⌊n c ⌋, where 1 < c < 2, can be locally approximated by sequences of the form ⌊nα + β⌋ in a very good way. Following this approach, we are led to an estimate of the differencewhich measures the deviation of the mean value of ϕ on the subsequence ⌊n c ⌋ from the expected value, by an expression involving exponential sums. As an application we prove that for 1 < c ≤ 1.42 the subsequence of the Thue-Morse sequence indexed by ⌊n c ⌋ attains both of its values with asymptotic density 1/2.