let $p$ be a real non-constant polynomial, prove that $\lim_{n \to \infty}\int_{0}^{1} e^{2\pi i n p(x)}dx=0$.

Question

let $p$ be a real non-constant polynomial, prove that $\lim_{n \to \infty}\int_{0}^{1} e^{2\pi i n p(x)}dx=0$.

The inside it bounded by $1$ and so we can use DCT, I just do not know how to show the inside converges to $0$.

Answer 1

It is enough to show that for any strictly monotonic function of class $C^1$ and any open interval $I\subset[0,1]$
we have $$\lim_{n\to +\infty}\int_{I} e^{in f(x)}\,dx = 0$$ due to the substitution $x=f^{-1}(u)$ and the Riemann-Lebesgue lemma.
Polynomials over compact intervals are $C^1$ and piecewise-monotonic.

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Numerical experiments seem to support the claim that $$ \int_{0}^{1}\exp\left(2\pi i n p(x)\right)\,dx = O\left(\frac{1}{n^{1/d}}\right) $$ where $d$ is the minimum degree of the monomials appearing in $p(x)-p(0)$.