The Mahler measure of the Rudin-Shapiro polynomials

Littlewood polynomials are polynomials with each of their coefficients in {-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin-Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials have been known before

Rational series and asymptotic expansion for linear homogeneous divide-and-conquer recurrences

Among all sequences that satisfy a divide-and-conquer recurrence, the sequences that are rational with respect to a numeration system are certainly the most immediate and most essential. Nevertheless, until recently they have not been studied from the asymptotic standpoint. We show how a mechanical process permits to compute their asymptotic expansion. It is based on linear algebra, with Jordan normal form, joint spectral radius, and dilation equations. The method is compared with the analytic number theory approach, based on Dirichlet series and residues, and new ways to compute the Fourier series of the periodic functions involved in the expansion are developed. The article comes with an extended bibliography

On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges

Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let $N(I,R_k-n)$ denote the number of zeros of $R_k-n$ in an interval $I := [\alpha,\beta] \subset [0,2\pi]$. Improving earlier results stated only for $I := [0,2\pi]$, in this paper we show that $$\frac{n|I|}{8\pi} - \frac{2}{\pi} (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}{\pi} + \frac{8}{\pi}(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every $I := [\alpha,\beta] \subset [0,2\pi]$, where $|I| = \beta-\alpha$ denotes the length of the interval $I$.Comment: 7 page

Dirichlet polynomials: some old and recent results, and their interplay in number theory

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and continue with some investigations concerning their supremum properties. Their random counterpart is considered in the last part of the paper, where a analysis of their supremum properties, based on methods of stochastic processes, is developed.Comment: 29 page

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization