Mixing of 3-term progressions in Quasirandom Groups

In this note, we show the mixing of three-term progressions (x, xg, xg^2) in every finite quasirandom groups, fully answering a question of Gowers. More precisely, we show that for any D-quasirandom group G and any three sets A_1, A_2, A_3 ⊂ G, we have |_x,y∼ G[ x ∈ A_1, xy ∈ A_2, xy^2 ∈ A_3] - ∏_i=1^3 _x∼ G[x ∈ A_i] | ≤(2/√(D))^1/4. Prior to this, Tao answered this question when the underlying quasirandom group is SL_d(𝔽_q). Subsequently, Peluse extended the result to all nonabelian finite simple groups. In this work, we show that a slight modification of Peluse's argument is sufficient to fully resolve Gower's quasirandom conjecture for 3-term progressions. Surprisingly, unlike the proofs of Tao and Peluse, our proof is elementary and only uses basic facts from nonabelian Fourier analysis.

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1 Introduction

In this note, we revisit a conjecture by Gowers [Gowers2008] about mixing of three term arithmetic progressions in quasirandom finite groups. Gowers initiated the study of quasirandom groups while refuting a conjecture of Babai and Sós [BabaiS1985] regarding the size of the largest product-free set in a given finite group. A finite group is said to be -quasirandom for a positive integer if all its non-trivial irreducible representations are at least -dimensional. The quasirandomness property of groups can be used to show that certain "objects" related to the group "mix" well. For instance, the quasirandomness of the group can be used to give an alternate (and weaker) proof [DavidoffSV-ramanujan] that the Ramanujan graphs of Lubotzky, Philips and Sarnak [LubotzkyPS1988] are expanders. Bourgain and Gambard [BourgainG2008] used quasirandomness to prove that certain other Cayley graphs were expanders.

Gowers proved that for any -quasirandom group and any three subsets satisfying , there exist such that . More generally, he proved that the number of such triples such that is at least provided . In other words the set of triples of the form mix well in a quasirandom group. Gowers’ proof of this result was the inspiration and the first step towards the recent optimal inapproximability result for satisfiable LIN over nonabelian groups [BhangaleK2021]. After proving the well-mixing of triples of the form in quasirandom groups, Gowers conjectured a similar statement for triples of the form . More precisely, he conjectured the following statement: Let be a -quasirandom group and such that , then

(1)

where the expression goes to zero as increases. The conjecture can be naturally extended to -term arithmetic progressions and product of functions for . However, in this note we will focus on the three term case.

For the specific case of 3-term progressions, Tao [Tao2013] proved the conjecture for the group for bounded using algebraic geometric machinery. In particular, he proved that the right-hand side expression in Eq. 1 can be bounded by when and for larger . Tao’s approach relied on algebraic geometry and was not amenable to other quasirandom groups. Later, Peluse [Peluse2018] proved the conjecture for all nonabelian finite simple groups. She used basic facts from nonabelian Fourier analysis to prove that the right-hand side expression in Eq. 1 can be bounded by where represents the set of irreducible unitary representation of and the dimension of the irreducible representation . This latter quantity is the Witten zeta function of the group minus one and can be bounded for simple finite quasirandom groups using a result due to Liebeck and Shalev [LiebeckS2004].

In this paper, we show that a slight variation of Peluse’s argument can be used to prove the conjecture for all quasirandom groups with better error parameters. More surprisingly, the proof stays completely elementary and short. Specifically, we prove the following statement:

Theorem 1.

Let be a -quasirandom finite group, i.e, its all non-trivial irreducible representations are at least -dimensional. Let such that then

2 Preliminaries

We begin by recalling some basic representation theory and nonabelian Fourier analysis. See the monograph by Diaconis [Diaconis1998, Chapter 2] for a more detailed treatment (with proofs).

We will be working with a finite group and complex-valued functions on

. All expectations will be with respect to the uniform distribution on

. The convolution between two function , denoted by , is defined as follows:

For any , the -norm of any function is defined as

For any element , the conjugacy class of , denoted by , refers to the set . Observe that the conjugacy classes form a partition of the group . A function is said to be a class function if it is constant on conjugacy classes.

For any we use . For any set , denotes the scaled density function . The scaling ensures that .

Given a complex vector space

, we denote the vector space of linear operators on by . This space is endowed with the following inner product and norm (usually referred to as the Hilbert-Schmidt norm):

This norm is known to be submultiplicative (i.e, ).

Representations and Characters:

A representation is a homomorphism from to the set of linear operators on for some finite-dimensional vector space over , i.e., for all , we have . The dimension of the representation , denoted by , is the dimension of the underlying -vector space . The character of a representation , denoted by , is defined as .

The representation satisfying for all is the trivial representation. A representation is said to reducible if there exists a non-trivial subpsace such that for all , we have . A representation is said to be irreducible otherwise. The set of all irreducible representations of (upto equivalences) is denoted by .

For every representation , there exists an inner product over such that every is unitary (i.e, for all and ). Hence, we might wlog. assume that all the representations we are considering are unitary.

The following are some well-known facts about representations and characters.

Proposition 2.
  1. The group is abelian iff for every irreducible representation in .

  2. For any finite group , .

  3. [orthogonality of characters] For any we have: .

Definition 3 (quasirandom groups).

A nonabelian group is said to be -quasirandom for some positive integer if all its non-trivial irreducible representations satisfy .

Nonabelian Fourier analysis:

Given a function and an irreducible representation

, the Fourier transform is defined as follows:

The following proposition summarizes the basic properties of Fourier transform that we will need.

Proposition 4.

For any , we have the following

  1. [Fourier transform of trivial representation]

  2. [Convolution]

  3. [Fourier inversion formula]

  4. [Parseval’s identity]

  5. [Fourier transfrom of class functions] For any class function , the Fourier transform satisfies

    for some constant . In other words, the Fourier transform is a scaling of the Identity operator .

The following claim (also used by Peluse [Peluse2018]) observes that the scaled density function has a very simple Fourier transform since it is a translate of the class function

Claim 5.

For any and we have:

where refers to the conjugacy class of . Moreover,

Proof.

We begin by observing that

On the other hand, as is a class function, we have for some constant . The constant can be determined by taking trace on either side of and noting that as follows:

Hence, and . Lastly we have,

(By unitariness of )

The key property of -quasirandom groups that we will be using is the following inequality due to Babai, Nikolov and Pyber, the proof of which we provide for the sake of completeness.

Lemma 6 ([BabaiNP2008]).

If is a -quasirandom group and such that either or is mean zero then

Proof.
(By submultiplicativity of norm)
(By mean zeroness)
(By -quasirandomness)

The following is a simple corrollary of Lemma 6.

Corollary 7.

If is -quasirandom; has zero mean and then

Proof.

Let . We have,

(By Cauchy-Schwarz inequality)
(By Lemma 6)
(Since ).

3 Proof of creftype 1

The following proposition is where we deviate from Peluse’s proof [Peluse2018]. We give an elementary proof for every quasirandom group while Peluse proved the same result for simple finite groups using the result of Liebeck and Shalev [LiebeckS2004] to bound the Witten zeta function for simple finite groups.

Proposition 8.

Let be a -quasirandom group. Let such that , and is the mean zero component of the function (i.e., ). Then

Proof.

Let us denote the expression on the L.H.S. as . We use simple manipulations and previously stated facts to simplify the expression.

(By Cauchy-Schwarz inequality)
(Since )
(By Cauchy Schwarz inequality)
(By Parseval’s identity & )
(By submultiplicativity of norm)
(By creftype 5)

Now using the fact that is uniformly distributed in for a fixed and a uniformly random in , we can simply the above expression as follows.

(By orthogonality of )

Finally, we use the fact that all the terms in the summation are non-negative and the group is a -quasirandom group.

(By Parseval’s identity)
(Because ).

The proof of this lemma is similar to the proof of the BNP inequality (Lemma 6). The key difference being that we have a complete characterization of the Fourier transform of from creftype 5 which we use to give a sharper bound. ∎

We are now ready to prove the main creftype 1. This part of the proof is similar to the corresponding expression that appears in the paper of Peluse [Peluse2018], which is in turn inspired by Tao’s adaptation of Gowers’ repeated Cauchy-Schwarzing trick to the nonebelian setting. We, however, present the entire proof for the sake of completeness.

Proof of creftype 1.

Let us denote the L.H.S. of the expression by . Without loss of generality we assume . Now we have,

(Change of variables: )
(Cauchy-Schwarz over ; and expansion )
(Change of variables: )
(Cauchy-Schwarz over ; ).

Now, using the following change of variables, , we get

We now separate the function from its the mean zero part as follows: Let where and .

(Using creftype 8 to bound the first expectation)
(Using )
(By creftype 7 and ).

References