Английская Википедия:Hua's lemma

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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, <math>\varepsilon</math> is a positive real number, and f a real function defined by

<math>f(\alpha)=\sum_{x=1}^N\exp(2\pi iP(x)\alpha),</math>

then

<math>\int_0^1|f(\alpha)|^\lambda d\alpha\ll_{P, \varepsilon} N^{\mu(\lambda)}</math>,

where <math>(\lambda,\mu(\lambda))</math> lies on a polygonal line with vertices

<math>(2^\nu,2^\nu-\nu+\varepsilon),\quad\nu=1,\ldots,k.</math>