Bourgain-Ruzsa Type Estimates for the Maximum and Minimum of Littlewood Cosine polynomials on the Real Line

We prove that $$\underset{t\in \mathbb{R}}{\max }{T}_n(t)\ge \frac{1}{60}{n}^{1\mathrm{/}3}\text{and}-\underset{t\in \mathbb{R}}{\min }{T}_n(t)\ge \frac{1}{60}{n}^{1\mathrm{/}3}$$ for every trigonometric polynomial $T_n$ of the form $$T_n(t)=\sum _{j=1}^n\cos (jt-{\theta }_j),{\theta }_j\in \mathbb{R},t\in \mathbb{R}\mathrm{.}$$