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, $\varepsilon$ is a positive real number, and f a real function defined by

$f(\alpha )=\sum _{x=1}^{N}\exp(2\pi iP(x)\alpha ),$

then

$\int _{0}^{1}|f(\alpha )|^{\lambda }d\alpha \ll _{P,\varepsilon }N^{\mu (\lambda )}$,

where $(\lambda ,\mu (\lambda ))$ lies on a polygonal line with vertices

$(2^{\nu },2^{\nu }-\nu +\varepsilon ),\quad \nu =1,\ldots ,k.$