On the real roots of generalized Thue-Morse polynomials

In this article we investigate real roots of real polynomials. By results of M. Kac [8-10] we know that a polynomial of degree n has on average (2/π)log n real zeros. See also results of Edelman and Kostlan [6] on the same subject. Some 10 years later Erdös and Offord [7] proved that the mean number of real roots of a random polynomial of degree n with coefficients ±1 is again (2/π) log n. This leads us to the following question: can we find sequences (α_i)_i∈ℕ with coefficients ±1 such that the corresponding polynomials Σⁿ_i=0α_iXⁱ have O(logn) real roots, and are these sequences random in some sense? We introduce generalized Thue-Morse sequences whose corresponding polynomials of large degree n have at least C log n real roots, where C is an explicit positive constant. Finally, we discuss the spectral measure of these sequences.