Hua's lemma

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, ε {\displaystyle \varepsilon } is a positive real number, and f a real function defined by

f ( α ) = ∑ x = 1 N exp ⁡ ( 2 π i P ( x ) α ) , {\displaystyle f(\alpha )=\sum _{x=1}^{N}\exp(2\pi iP(x)\alpha ),}

then

∫ 0 1 | f ( α ) | λ d α ≪ P , ε N μ ( λ ) {\displaystyle \int _{0}^{1}|f(\alpha )|^{\lambda }d\alpha \ll _{P,\varepsilon }N^{\mu (\lambda )}} ,

where ( λ , μ ( λ ) ) {\displaystyle (\lambda ,\mu (\lambda ))} lies on a polygonal line with vertices

( 2 ν , 2 ν − ν + ε ) , ν = 1 , … , k . {\displaystyle (2^{\nu },2^{\nu }-\nu +\varepsilon ),\quad \nu =1,\ldots ,k.}

References