Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval

We consider polynomials $p_n^{\omega}(x)$ that are orthogonal with respect to the oscillatory weight $w(x)=e^{i\omega x}$ on $[-1,1]$, where $\omega>0$ is a real parameter. A first analysis of $p_n^{\omega}(x)$ for large values of $\omega$ was carried out in connection with complex Gaussian quadrature rules with uniform good properties in $\omega$. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of $p_n^{\omega}(x)$ in the complex plane as $n\to\infty$. The parameter $\omega$ grows with $n$ linearly. The tools used are logarithmic potential theory and the $S$-property, together with the Riemann--Hilbert formulation and the Deift--Zhou steepest descent method.