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Bhattacharyya, Arnab

doi:10.1145/2556663.2556678

Cited by 8 publications

References 72 publications

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Testing properties of functions on finite groups

Oono

Yoshida

2016

Random Struct Algorithms

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ABSTRACT:We study testing properties of functions on finite groups. First we consider functions of the form f : G → C, where G is a finite group. We show that conjugate invariance, homomorphism, and the property of being proportional to an irreducible character is testable with a constant number of queries to f , where a character is a crucial notion in representation theory. Our proof relies on representation theory and harmonic analysis on finite groups. Next we consider functions of the form

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Testing properties of functions on finite groups

Oono

Yoshida

2016

Random Struct Algorithms

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Sunflowers and Testing Triangle-Freeness of Functions

Haviv

Xie

2016

comput. complex.

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A function f : F n 2 → {0, 1} is triangle-free if there are no x 1 , x 2 , x 3 ∈ F n 2 satisfying x 1 + x 2 + x 3 = 0 and f (x 1 ) = f (x 2 ) = f (x 3 ) = 1. In testing triangle-freeness, the goal is to distinguish with high probability triangle-free functions from those that are ε-far from being triangle-free. It was shown by Green that the query complexity of the canonical tester for the problem is upper bounded by a function that depends only on ε (GAFA, 2005), however the best known upper bound is a tower type function of 1/ε. The best known lower bound on the query complexity of the canonical tester is 1/ε 13.239 (Fu and Kleinberg, RANDOM, 2014).In this work we introduce a new approach to proving lower bounds on the query complexity of triangle-freeness. We relate the problem to combinatorial questions on collections of vectors in Z n D and to sunflower conjectures studied by Alon, Shpilka, and Umans (Comput. Complex., 2013). The relations yield that a refutation of the Weak Sunflower Conjecture over Z 4 implies a super-polynomial lower bound on the query complexity of the canonical tester for triangle-freeness. Our results are extended to testing k-cycle-freeness of functions with domain F n p for every k ≥ 3 and a prime p. In addition, we generalize the lower bound of Fu and Kleinberg to k-cycle-freeness for k ≥ 4 by generalizing the construction of uniquely solvable puzzles due to Coppersmith and Winograd (J. Symbolic Comput., 1990).

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A tight bound for Green's arithmetic triangle removal lemma in vector spaces

Fox

Lovász

2017

Advances in Mathematics

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Let p be a fixed prime. A triangle in F n p is an ordered triple (x, y, z) of points satisfying x + y + z = 0. Let N = p n = |F n p |. Green proved an arithmetic triangle removal lemma which says that for every ǫ > 0 and prime p, there is a δ > 0 such that if X, Y, Z ⊂ F n p and the number of triangles in X × Y × Z is at most δN 2 , then we can delete ǫN elements from X, Y , and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1/δ can be taken to be an exponential tower of twos of height logarithmic in 1/ǫ.We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in F n p . We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a explicit number C p such that we may take δ = (ǫ/3) Cp , and we must have δ ≤ ǫ Cp−o(1) . In particular, C 2 = 1 + 1/(5/3 − log 2 3) ≈ 13.239, and C 3 = 1 + 1/c 3 with c 3 = 1 − log b log 3 , b = a −2/3 + a 1/3 + a 4/3 , and a = √ 33−1 8, which gives C 3 ≈ 13.901. The proof uses the essentially sharp bound on multicolored sum-free sets due to work of Kleinberg-SawinSpeyer, Norin, and Pebody, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-GrochowNaslund-Sawin-Umans, and Alon.

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Guest column

Cited by 8 publications

References 72 publications

Testing properties of functions on finite groups

Testing properties of functions on finite groups

Sunflowers and Testing Triangle-Freeness of Functions

A tight bound for Green's arithmetic triangle removal lemma in vector spaces

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