Hua's lemma

In mathematics, Hua's lemma, ["On Waring's problem", Quarterly Journal of Mathematics, 9, pages 199-202] 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

:$f(alpha)=sum\_\{x=1\}^Nexp(2pi\; iP(x)alpha),$

then

:$int\_0^1|f(alpha)|^lambda\; dalphale\; CN^\{mu(lambda)\}$,

where $(lambda,mu(lambda))$ lies on a polygonal line with vertices

:$(2^\; u,2^\; u-\; u+varepsilon),quad\; u=1,ldots,k,$

and "C" is some positive number only depending on the coefficients of "P" and $varepsilon$.

References