Hua's lemma

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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\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

  1. ^ On Waring's problem, Quarterly Journal of Mathematics, 9, pages 199-202