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Given a function f on (F_2_^n, we study the following problem. What is the largest affine subspace U such that when restricted to U, all the non-trivial Fourier coefficients of f are very small? For the natural class of bounded Fourier degree d functions f : (F_2)^n → [−1, 1], we show that there exists an affine subspace of dimension at least ˜Ω (n^(1/d!) k^(−2)), wherein all of f ’s nontrivial Fourier coefficients become smaller than 2^(−k) . To complement this result, we show the existence of degree d functions with coefficients larger than 2^(−d log n) when restricted to any affine subspace of dimension larger than Ω(dn^(1/(d−1))). In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of (F_2)^n that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.
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Searching for Regularity in Bounded Functions
This work studies the problem of finding a large affine subspace over the field F_2 such that a bounded function's nontrivial Fourier coefficients become small. We show that for any function f from F_2^n to [-1,1] with Fourier degree d, there exists an affine subspace of dimension at least roughly n^{1/d!}k^{-2}), wherein all of f's nontrivial Fourier coefficients become smaller than 2^{-k}. To complement this result, we show the existence of degree d functions with coefficients larger than 2^{-d log n} on any subspace of dimension larger than Omega(dn^{1/(d-1)}). In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of $$\F_2^n$$ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers/extractors.
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- Award ID(s):
- 2006359
- PAR ID:
- 10343605
- Date Published:
- Journal Name:
- Electronic colloquium on computational complexity
- Volume:
- TR22
- Issue:
- 109
- ISSN:
- 1433-8092
- Page Range / eLocation ID:
- 1-26
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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