A sharp bound for the Stein-Wainger oscillatory integral

Let $\mathcal{P}_d$ denote the space of all real polynomials of degree at most $d$. It is an old result of Stein and Wainger that

$\sup_ {P\in\mathcal{P}_d} \bigg\vert p.v.\int_{\mathbb{R}} {e^{iP(t)}\frac{dt}{t}} \bigg\vert\leq C_d$

for some constant $C_d$ depending only on $d$. On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is $\log d$. We prove that

$\sup_ {P\in\mathcal{P}_d}\bigg\vert p.v. \int_{\mathbb{R}}{e^{iP(t)}\frac{dt}{t}}\bigg\vert\sim \log{d}.$