Seems like Ben Green and (Fields Medalist) Terry Tao are into the analysis of boolean functions more and more these days. Check out Terry's recent blog post on Freiman's theorem in $\mathbb{F}_2^n$, AKA $\{-1,1\}^n$, and Ben's earlier guest post on the same blog on the Polynomial Freiman-Ruzsa conjecture. Their latest two papers (mentioned in Terry's post) have all of our favorites -- the boolean cube, Fourier analysis, monotone functions, shifting, quasirandomness, linear threshold functions.

If only I understood them...

As usual, they look impeccably written, so it's just a matter of putting in the time to read them. If any of you wants to do this and then explain them to me, that would be great :)

For a warmup, I always recommend Ben Green's terrific survey article, Finite Field Models In Additive Combinatorics. The main difference between their world and ours is that we obsess over the low-degree Fourier coefficients, treating them as most notable. On the other hand, they are perfectly egalitarian about $\hat{f}(S)$ across $S$; from their arithmetic point of view, all parity functions are created equal.

## Tuesday, March 27, 2007

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