Hua's lemma

In mathematics, Hua's lemma,^[1] named for Hua Luogeng, 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

$\int_0^1|f(\alpha)|^\lambda d\alpha\le CN^{\mu(\lambda)}$ ,

where $(λ,μ(λ))$ lies on a polygonal line with vertices

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

and C is some positive number only depending on the coefficients of P and $\varepsilon$ .

[edit] References