Quasirandom Group Actions

Gill, Nick

doi:10.1017/fms.2016.8

Cited by 8 publications

(11 citation statements)

References 27 publications

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“…The Frobenius theorem [10] on representations of G gives the following bound (see [13], [15], [27]) for the convolution of any functions f 1 and f 2 with zero mean:…”

Section: Proof Of Lemmamentioning

confidence: 99%

On growth rate in $SL_2(\mathbf{F}_p)$, the affine group and sum-product type implications

Rudnev¹,

Shkredov²

2018

Preprint

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This paper aims to study in more depth the relation between growth in matrix groups SL 2 (F) and Aff(F) over a field F by multiplication and geometric incidence estimates, associated with the sum-product phenomenon over F. It presents streamlined proofs of Helfgott's theorems on growth in the F p -case, which avoid sum-product estimates. For SL 2 (F p ), for sets exceeding in size some absolute constant, we improve the lower bound 1 1512 for the growth exponent, due to Kowalski, to 1 20 . For the affine group we fetch a sharp theorem of Szőnyi on the number of directions, determined by a point set in F 2 p . We then focus on Aff(F) and present a new incidence bound between a set of points and a set of lines in F 2 , which explicitly depends on the energy of the set of lines as affine transformations under composition. This bound, strong when the number of lines is considerably smaller than the number of points, yields generalisations of structural theorems of Elekes and Murphy on rich lines in grids.In the special case when the set of lines is also a grid -relating back to sum-products -we use growth in Aff(R) to obtain a subthreshold estimate on the energy of the set of lines. This yields a unified way to break the ice in various threshold sum-product type energy inequalities. We show this in applications to energy estimates, corresponding to sets A(A + A), A + AA (also embracing asymmetric versions) as well as A + B when A has small multiplicative doubling and |A| |B| |A| 1+o(1) . * The first author is supported by the Leverhulme Trust Grant RPG-2017-371. 1 20 .1 We thank H. Helfgott for pointing out that owing to the structure of the proof of Theorem 2, the outcome of Corollary 1 is almost as strong as if we had K ≫ |A|

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“…The Frobenius theorem [10] on representations of G gives the following bound (see [13], [15], [27]) for the convolution of any functions f 1 and f 2 with zero mean:…”

Section: Proof Of Lemmamentioning

confidence: 99%

On growth rate in $SL_2(\mathbf{F}_p)$, the affine group and sum-product type implications

Rudnev¹,

Shkredov²

2018

Preprint

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“…Let us mention a well-known lemma (see [9], [21], [44] and other papers) on convolutions in SL 2 (F p ) which follows from the well-known Frobenius Theorem [20] on representations of SL 2 (F p ). For the sake of completeness we add the proof of this lemma in the Appendix.…”

Section: Imentioning

confidence: 99%

“…By the famous Frobenius result [20] the dimension of all nontrivial irreducible representations of SL 2 (F p ) is at least (p − 1)/2. It follows that for any eigenfunction v, v = u the multiplicity of the correspondent eigenvalue is at least (p − 1)/2 (see details in [44], [10], [21]). Hence in view of (151), we obtain λ 1 ≤ 2p f 2 .…”

Section: On Gl 2 (F P )-Actionsmentioning

confidence: 99%

“…In the proofs we use usual incidence theorems in F p , see [42], [53] and other papers, as well as the growth result in SL 2 (F p ) due to Helfgott [22]. So, our another aim is to obtain some new applications (also, see recent papers [32], [31] where other applications were found) of classical graph (group) expansion phenomena, see [9], [21], [18], [22], [44] and others.…”

Section: Introductionmentioning

confidence: 99%

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On asymptotic formulae in some sum-product questions

Shkredov¹

2018

Preprint

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In this paper we obtain a series of asymptotic formulae in the sum-product phenomena over the prime field F p . In the proofs we use usual incidence theorems in F p , as well as the growth result in SL 2 (F p ) due to Helfgott. Here some of our applications:• a new bound for the number of the solutions to the equationis an arbitrary subset of F p , • a new effective bound for multilinear exponential sums of Bourgain, • an asymptotic analogue of the Balog-Wooley decomposition theorem, • growth of p 1 (b) + 1/(a + p 2 (b)), where a, b runs over two subsets of F p , p 1 , p 2 ∈ F p [x] are two nonconstant polynomials,• new bounds for some exponential sums with multiplicative and additive characters.

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“…Thus but for the case of sufficiently large sets, where one can use Fourier analysis and representation theory to describe quasirandom group actions (see e.g. [3] and references therein), the only way to connect this with the energy of a set of transformations has been via the non-commutative Balog-Szemerédi-Gowers Theorem.…”

Section: Introductionmentioning

confidence: 99%

An Energy Bound in the Affine Group

Petridis,

Roche-Newton,

Rudnev

et al. 2019

Preprint

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We prove a nontrivial energy bound for a finite set of affine transformations over a field F and discuss a number of implications. These include new bounds on growth in the affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P , there may be at most one line l, meeting the set of lines defined by P in ≪ |P | points.

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Quasirandom Group Actions

Cited by 8 publications

References 27 publications

On growth rate in $SL_2(\mathbf{F}_p)$, the affine group and sum-product type implications

On growth rate in $SL_2(\mathbf{F}_p)$, the affine group and sum-product type implications

On asymptotic formulae in some sum-product questions

An Energy Bound in the Affine Group

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