Yitzhak Katznelson

The influence of variables in product spaces

AbstractLetX be a probability space and letf: Xn → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byIf(k), as follows: Foru=(u1,u2,…,un−1) ∈Xn−1 consider the setlk(u)={(u1,u2,...,uk−1,t,uk,…,un−1):t ∈X}.

Appendix on return-time sequences

I_f (k) = \Pr (u \in X^{n - 1} :f is not constant on l_k (u)).

The two sides of a fourier-stieltjes transform and almost idempotent measures

More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byIf(S), be the probability that assigning values to the variables not inS at random, the value off is undetermined. Theorem 1:There is an absolute constant c1so that for every function f: Xn → {0, 1},with Pr(f−1(1))=p≤1/2,there is a variable k so that

On certain homomorphisms of quotients of group algebras

I_f (k) \geqslant c_1 p\frac{{\log n}}{n}.

Sets of uniqueness and multiplicity forl p,α

Theorem 2:For every f: Xn → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c2(ε)n/logn so that If (S)≥1−ε.These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}.

An Introduction to Harmonic Analysis

We prove a theorem about idempotents in compact semigroups. This theorem gives a new proof of van der Waerden’s theorem on arithmetic progressions as well as the Hales-Jewett theorem. It also gives an infinitary version of the Hales-Jewett theorem which includes results of T. J. Carlson and S. G. Simpson.