Sums of Linear Transformations in Higher Dimensions

Mudgal, Akshat

doi:10.1093/qmath/haz006

Cited by 4 publications

(4 citation statements)

References 16 publications

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“…In this section we use Balog-Szemerédi-Gowers [2,12] in combination with a structural result of Mudgal [20] which uses the general Freiman-type theorem of Green and Ruzsa [13]. Our aim in this section is to demonstrate that our point set has large intersection with a line.…”

Section: Additive Case: Line Structurementioning

confidence: 99%

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A Structural Theorem for Sets With Few Triangles

Mansfield¹,

Passant²

2022

Preprint

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We show that if a finite point set P has the fewest congruence classes of triangles possible, up to a constant M , then at least one of the following holds.• There is a polynomially-rich line l through P and a positive proportion of P is contained on lines parallel to l.• There is a circle γ which contains a positive proportion of P . Under slightly stronger assumptions, this establishes a conjecture of Erdős. We use the result of Petridis-Roche-Newton-Rudnev-Warren on the structure of the affine group combined with classical results from additive combinatorics.

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Section: Additive Case: Line Structurementioning

confidence: 99%

“…The additive case we require the application of Balog-Szemerédi-Gowers and a result of Mudgal [20] that relies on the Freiman-type result of Green-Ruzsa [13]. We summarise steps 2 to 4 in the following Theorem.…”

Section: Introductionmentioning

confidence: 99%

A Structural Theorem for Sets With Few Triangles

Mansfield¹,

Passant²

2022

Preprint

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“…In particular, one may consider lower bounds for cardinalities of sets of the from A + L (A), where A ⊆ R d with d ≥ 2, and L is some invertible linear transformation. The case when L is a particular type of integer dilation was studied by Balog and Shakan in [1], which was then sharpened and generalised by the author in [4]. Furthermore, the case of L being a rotation matrix in R 2 was considered in [4], and we refer the reader to this paper for a more detailed exposition on this problem.…”

Section: Introductionmentioning

confidence: 99%

“…The case when L is a particular type of integer dilation was studied by Balog and Shakan in [1], which was then sharpened and generalised by the author in [4]. Furthermore, the case of L being a rotation matrix in R 2 was considered in [4], and we refer the reader to this paper for a more detailed exposition on this problem.…”

Section: Introductionmentioning

confidence: 99%

Difference sets in higher dimensions

Mudgal

2020

Math. Proc. Camb. Phil. Soc.

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Let d ≥ 3 be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have $$\begin{equation*} |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\end{equation*}$$ where δ > 0 is an absolute constant only depending on d. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.

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A Structural Theorem for Sets with Few Triangles

Mansfield,

Passant

2023

Combinatorica

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We show that if a finite point set $$P\subseteq {\mathbb {R}}^2$$ P ⊆ R 2 has the fewest congruence classes of triangles possible, up to a constant M, then at least one of the following holds. There is a $$\sigma >0$$ σ > 0 and a line l which contains $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. Further, a positive proportion of P is covered by lines parallel to l each containing $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. There is a circle $$\gamma $$ γ which contains a positive proportion of P. This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.

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Sums of Linear Transformations in Higher Dimensions

Cited by 4 publications

References 16 publications

A Structural Theorem for Sets With Few Triangles

A Structural Theorem for Sets With Few Triangles

Difference sets in higher dimensions

A Structural Theorem for Sets with Few Triangles

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