For and every , the displayed estimate bounds a mean value of a polynomial exponential sum. For cubes its eighth-moment bound is . It complements pointwise Weyl differencing estimates: a ninth moment over a region is bounded by its supremum there times this eighth moment. Leiden lecture notes, Section 8.1 give the cubic formulation.
Write and . Character orthogonality gives . For , fix the difference . If , there are choices for the two equal pairs. If , any solution has ; fixing this divisor gives a quadratic equation for , hence at most two choices. The subpower bound for the divisor function yields .
The twice-differenced inequality in Weyl differencing reads , with counting triples for which and all four shifted arguments are in . Thus , and for : the product divides , and the remaining equation determines . The Fourier coefficients of are nonnegative counts, with and . Multiply the inequality by and integrate. It follows that , after choosing the divisor-bound exponents small enough.
Articles by others on the same topic
Hua's lemma is a result in number theory, particularly in the area of additive number theory, often associated with the work of the Chinese mathematician Hua Luogeng. It generally pertains to the distribution of integers and can be used in problems related to additive representations or counting problems. The lemma can be formulated in terms of a sum over integers, usually involving counting the number of ways an integer can be expressed as a sum of a fixed number of integers from a specific set.