Hua's lemma explained

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

then

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

where

( λ ,\mu( λ )) lies on a polygonal line with vertices( 2 ^{\nu,2} ^{\nu-\nu+\varepsilon), \nu=1,\ldots,k.}

Notes and References

1. Hua Loo-keng . Hua Luogeng . 1938 . On Waring's problem . Quarterly Journal of Mathematics . 9 . 1 . 199–202 . 10.1093/qmath/os-9.1.199 . free .