Hua's lemma

In mathematics, Hua's lemma,^[1] 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

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

References

1. Hua Loo-keng (1938). "On Waring's problem". Quarterly Journal of Mathematics 9 (1): 199–202. doi:10.1093/qmath/os-9.1.199.