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),


\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



  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.