In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norm on functions on a finite group or group-like object which are used in the study of arithmetic progressions in the group. It is named after Timothy Gowers, who introduced it in his work on Szemerédi's theorem.
Let f be a complex-valued function on a finite Abelian group G and let J denote complex conjugation. The Gowers d-norm is
The inverse conjecture for these norms is the statement that if f has L-infinity norm (uniform norm in the usual sense) equal to 1 then the Gowers s-norm is bounded above by 1, with equality if and only if f is of the form exp(2πi g) with g a polynomial of degree at most s. This can be interpreted as saying that the Gowers norm is controlled by polynomial phases.
The inverse conjecture holds for vector spaces over a finite field. However, for cyclic groups Z/N this is not so, and the class of polynomial phases has to be extended to control the norm.
- Tao, Terence (2012). Higher order Fourier analysis. Graduate Studies in Mathematics 142. Providence, RI: American Mathematical Society. ISBN 978-0-8218-8986-2. Zbl pre06110460.
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