Hua's lemma

In mathematics, Hua's lemma, named for Hua Loo-keng, is an estimate for exponential sums. It states that if P is an integral-valued polynomial of degree k, [math]\displaystyle{ \varepsilon }[/math] is a positive real number, and f a real function defined by

[math]\displaystyle{ f(\alpha)=\sum_{x=1}^N\exp(2\pi iP(x)\alpha), }[/math]

then

[math]\displaystyle{ \int_0^1|f(\alpha)|^\lambda d\alpha\ll_{P, \varepsilon} N^{\mu(\lambda)} }[/math],

where [math]\displaystyle{ (\lambda,\mu(\lambda)) }[/math] lies on a polygonal line with vertices

[math]\displaystyle{ (2^\nu,2^\nu-\nu+\varepsilon),\quad\nu=1,\ldots,k. }[/math]