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, $\epsilon$ 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

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

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

$(2^{\nu },2^{\nu }-\nu +\epsilon ),\nu =1,\dots ,k.$