A further restriction on translation-invariant spaces would be to consider affine-invariant spaces (space that are invariant under affine transformations of the domain). Polynomials appear here quite naturally and are the only affine-invariant spaces when the domain is a vector space over a *prime* field. But if the domain is not a vector space over a prime field, then spaces other than polynomials do exist, and indeed can be potentially be used as substitutes for polynomials in the polynomial method. One example (the only one I am aware of) where this gives some improvement is our work (with Guo and Kopparty) on Lifted codes which improve known bounds on Nikodym sets over fields of small characteristic. See http://arxiv.org/pdf/1208.5413v2.pdf (Theorem 1.6).

]]>“..a strong conviction that they always result from a fairly systematic process — and that the opposite impression, that some ideas are incredible bolts from the blue that require “genius” or “sudden inspiration” to find, is an illusion that results from the way mathematicians present their proofs after they have discovered them.”

Hmm, it will be quite exciting to discuss it. I do not think the issue is how proofs are presented. Sudden developments are not in contradiction with systematic processes (just think about graph processes.) And also there are, from time to time, inspirational surprising sharp turns.

“The third thing I’d like to do reflects another interest of mine, which is avoiding duplication of effort. I’ve spent a little time thinking about whether there is a cheap way of getting a Behrend-type bound for Roth’s theorem out of these ideas (and I’m not the only one). Although I wasn’t expecting the answer to be yes, I think there is some value in publicizing some of the dead ends I’ve come across.”

This is also a very interesting topic. Duplicating efforts and certainly competition can be beneficial, but presenting or hearing (sometimes) paths of attacks that failed can be interesting and of some value.

]]>Many thanks for this very useful comment.

]]>Ah that’s interesting.

]]>The classification of ideals depends on the structure of the ring. For functions on F_3^n over a finite field of characteristic not 3, the ring is a product of fields corresponding to characters, and any ideal just is a sum of fields corresponding to some subset of the characters – this is just Fourier analysis.

In characteristic 3, the ring is generated by the translation operators x_i =T_{e_i}-1 in the directions of a basis, each of which satisfies (T_{e_i}-1)^3 = 0, so the ring is F_3[x_1,…,x_n]/(x_1^3, …,x_n^3) – a nilpotent ring. This ring has many ideals which don’t really have a nice classification. The space of low degree polynomials corresponds, dually, to high degree polynomials in the x_i. You could restrict attention to special ideals, like ideals generated by polynomials – these correspond to the spaces of polynomials with all monomials appearing having multidegree in a certain downward-closed set of tuples of natural numbers.

However I think you can show that all spaces of functions that have the key property needed for this proof are quite close to the space of low degree polynomials.

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