On the Fourier spectrum of symmetric Boolean functions


We study the following question

What is the smallest t such that every symmetric boolean function on κ variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t?

We exclude the constant functions for which there is no such t and the parity functions for which t has to be κ. Let τ (κ) be the smallest such t. Our main result is that for large κ, τ (κ)≤4κ/logκ.

The motivation for our work is to understand the complexity of learning symmetric juntas. A κ-junta is a boolean function of n variables that depends only on an unknown subset of κ variables. A symmetric κ-junta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric κ-juntas, in the uniform PAC learning model, in time n o(κ). This improves on a result of Mossel, O’Donnell and Servedio in [16], who show that symmetric κ-juntas can be learned in time n 2κ/3.


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Corresponding author

Correspondence to Evangelos Markakis.

Additional information

This work was done when all authors were at the Georgia Institute of Technology and it is based on the preliminary versions [14] and [11].

Partially supported by European Commission IHP Network HARP (Harmonic Analysis and Related Problems), Contract Number: HPRN-CT-2001-00273 — HARP, and by grant INTAS 03-51-5070 (2004) (Analytical and Combinatorial Methods in Number Theory and Geometry).

Research supported by NSF grant CCF-0431023.

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Kolountzakis, M.N., Lipton, R.J., Markakis, E. et al. On the Fourier spectrum of symmetric Boolean functions. Combinatorica 29, 363–387 (2009). https://doi.org/10.1007/s00493-009-2310-z

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Mathematics Subject Classification (2000)

  • 42B05
  • 68Q32