Gowers Norm, Function Limits, and Parameter Estimation

Let {fi : Fip → {0, 1}} be a sequence of functions, where p is a fixed prime and Fp is the finite field of order p. The limit of the sequence can be syntactically defined using the notion of ultralimit. Inspired by the Gowers norm, we introduce a metric over limits of function sequences, and study properties of it. One application of this metric is that it provides a characterization of affine-invariant parameters of functions that are constant-query estimable. Using this characterization, we provide (alternative) proofs of the constant-query testability of several affine-invariant properties, including low-degree polynomials. ∗Supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 26730009), MEXT Grant-in-Aid for Scientific Research on Innovative Areas (24106001), and JST, ERATO, Kawarabayashi Large Graph Project.