1. On the Fourier Tails of Bounded Functions over the Discrete Cube Irit Dinur, Ehud Friedgut, and Ryan O’Donnell Joint work with Guy Kindler Microsoft Research

2. Fourier Analysis

3.  Fourier representation: can be written as a multilinear polynomial  is called the S Fourier coefficient of f.

4. Fourier Analysis  Fourier representation: can be written as a multilinear polynomial  is called the S Fourier coefficient of f.  Many structural properties of f can be inferred from its Fourier representation.  Useful in: hardness of approximation, circuit lower bounds, threshold phenomena, metric embeddings, algorithms, learning, communication complexity, complexity,…

5. Boolean vs. Bounded functions  Often one needs to study averages of Boolean functions.  Question: which properties persist for bounded functions?  Our initial motivation: coloring.  Ideas used in [KO 05] and [ABHKS 05].

6. What next: What next: Some technical background Some symmetry breaking phenomena for Boolean functions Main theorem: symmetry breaking for bounded functions Something about the proof.

7. On weights and tails  k-tail of f :  Low-degree part of f :  Weight:  k-tail weight:  Dinstance: Parseval’s identity.

8.  A J -junta: a function f that depends on at most J coordinates.  Often: having small k-tail weight implies f is junta-ish.  f is an ( ,J)-junta if 9 a J junta g such that  [FKN 02] ! f is an (O(  ),1)-junta.  [B 02] ! f is an (0.001,100 k )-junta.  For majority, the weight of the k-tail is. On Juntas and tails Symmetry breaking.

9.  A J -junta: a function f that depends on at most J coordinates.  Often: having small k-tail weight implies f is junta-ish.  f is an ( ,J)-junta if 9 a J junta g such that  [FKN 02] ! f is an (O(  ),1)-junta.  [B 02] ! f is an (0.001,100 k )-junta.  For majority, the weight of the k-tail is. Tails of bounded functions

10.  Is a threshold for k-tail bounded function?  No:  We have symmetric f with  Does there really exist a threshold ??  Theorem: If then it is an -junta.

11. what’s next: what’s next: Some technical background Some symmetry breaking phenomena for Boolean functions Main theorem: symmetry breaking for bounded functions Something about the proof.

12. Proof idea: use large deviations  Theorem: If then it is an -junta.  Idea: If f <k is smeared over many coordinates then it must obtain large values. So f  k must also obtain large values, and therefore have large weight.  We need a lower-bound on large deviations for low-degree functions.

13. Large deviation lower bounds  Linear case (folklore):,, and for all i. Then  Linear case (folklore):,, and for all i. Then  Main lemma:,, and for all i. Then

14. Vague idea of the proof

15. Conclusions and questions  Bounded functions do show symmetry-breaking phenomena.  This happens for different reasons and parameter-range than in the Boolean case.  Is there a generalization of Boolean functions where the same symmetry-breaking phenomena hold?  Get other bounded-case analogues for Boolean results.