L^\infty -estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization

On the real roots of generalized Thue-Morse polynomials

In this article we investigate real roots of real polynomials. By results of M. Kac 8, 9, 10] we know that a polynomial of degree n has in average 2 log n real zeros. See also results of Edelman and Kostlan 5] on the same subject. Some 10 years later Erdds and OOord 7] proved that the mean number of real roots of a random polynomial of degree n with coeecients 1 is again 2 log n: This leads us ...

Generalized Thue-Morse Sequences of Squares

We consider compact group generalizations T (n) of the Thue-Morse sequence and prove that the subsequence T (n) is uniformly distributed with respect to a measure ν that is absolutely continuous with respect to the Haar measure. The proof is based on a proper generalization of the Fourier based method of Mauduit and Rivat in their study of the sum-of-digits function of squares to group represen...

Integral Geometry and Real Zeros of Thue-Morse Polynomials

Factor frequencies in generalized Thue-Morse words

L'ubomíra Balková We describe factor frequencies of the generalized Thue–Morse word t b,m defined for b ≥ 2, m ≥ 1, b, m ∈ N, as the fixed point starting in 0 of the morphism ϕ b,m (k) = k(k + 1). .. (k + b − 1), where k ∈ {0, 1,. .. , m − 1} and where the letters are expressed modulo m. We use the result of Frid [4] and the study of generalized Thue–Morse words by Starosta [6].

Newman's phenomenon for generalized Thue-Morse sequences

Let tj = (−1)s(j) be the Thue-Morse sequence with s(j) denoting the sum of the digits in the binary expansion of j. A well-known result of Newman [10] says that t0 + t3 + t6 + · · ·+ t3k > 0 for all k ≥ 0. In the first part of the paper we show that t1 + t4 + t7 + · · ·+ t3k+1 < 0 and t2 + t5 + t8 + · · ·+ t3k+2 ≤ 0 for k ≥ 0, where equality is characterized by means of an automaton. This sharp...