O. Roche-Newton

activity

20112026

most citedNew results on sum-product type growth over fields

52 citations · 127 across the 37 of their papers we have counts for

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44 papers · 1 filter

math.CO2026

Growth beyond exponent $3/2$ for convexity and iterated sum sets

Michalis Kokkinos, Oliver Roche-Newton

We prove that the bound \[ \max \{ |16A|,|16f(A)| \} \gg_m |A|^{\frac{3}{2}+\frac{1}{162}} \] holds for any polynomial $f$ with degree $m \geq 2$ and any finite $A \subset \mathbb…

math.CO2026

Geometric Sidon Problems

Felix Christian Clemen, Jakob Führer, Oliver Roche-Newton

This paper considers geometric problems of the following type: given a point set $P \subset R^{2}$ , one seeks a large subset avoiding a prescribed geometric configuration. Ou…

math.CO2026

The minimum degree question for the Maker Breaker Domination Game

Jakob Führer, Georg Grasegger, Paul Hametner +1

The Maker Breaker Domination Game is a two player game played on a graph $G$ in which the players take turns to claim a vertex from the graph. The aim of the Dominator is to claim…

math.CO2023

Large Convex sets in Difference sets

Krishnendu Bhowmick, Ben Lund, Oliver Roche-Newton

We give a construction of a convex set $A \subset R$ with cardinality $n$ such that $A - A$ contains a convex subset with cardinality $Ω (n^{2})$ . We also consider the following…

math.CO2023

Local Differences Determined by Convex sets

Krishnendu Bhowmick, Miriam Patry, Oliver Roche-Newton

This paper introduces a new problem concerning additive properties of convex sets. Let $S = {s_{1} < \dots < s_{n}}$ be a set of real numbers and let $D_i(S)= \{s_x-s_y: 1 \leq x-y \l…

math.CO2023★ 2 cited

A better than $3/2$ exponent for iterated sums and products over $R$

Oliver Roche-Newton

In this paper, we prove that the bound \[ \max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}-o(1)} \] holds for all $A \subset R$ , and for all convex func…

math.CO2022

Counting arcs in $F_{q}^{2}$

Krishnendu Bhowmick, Oliver Roche-Newton

An arc in $F_{q}^{2}$ is a set $P \subset F_{q}^{2}$ such that no three points of $P$ are collinear. We use the method of hypergraph containers to prove several counting re…

math.CO2022

A convex set with a rich difference

Oliver Roche-Newton, Audie Warren

We construct a convex set $A$ with cardinality $2 n$ and with the property that an element of the difference set $A - A$ can be represented in $n$ different ways. We also show that th…

math.CO2022★ 2 cited

Convexity, Squeezing, and the Elekes-Szabó Theorem

Oliver Roche-Newton, Elaine Wong

This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing p…

math.CO2021★ 1 cited

Convexity, Superquadratic Growth, and Dot Products

Brandon Hanson, Oliver Roche-Newton, Steven Senger

Let $P \subset R^{2}$ be a point set with cardinality $N$ . We give an improved bound for the number of dot products determined by $P$ , proving that, \[ |\{ p \cdot q :p,q \in…

math.CO2021

Additive and multiplicative Sidon sets

Oliver Roche-Newton, Audie Warren

We give a construction of a set $A \subset N$ such that any subset $A^{'} \subset A$ with $∣ A^{'} ∣ ≫ ∣ A ∣^{2/3}$ is neither an additive nor multiplicative Sidon set. In doing so…

math.CO2020

The Elekes-Szabó Problem and the Uniformity Conjecture

Mehdi Makhul, Oliver Roche-Newton, Sophie Stevens +1

In this paper we give a conditional improvement to the Elekes-Szabó problem over the rationals, assuming the Uniformity Conjecture. Our main result states that for $F\in \mathbb{Q}…

math.CO2020

Sums, products and dilates on sparse graphs

Oliver Roche-Newton

Let $A \subset R$ and $G \subset A \times A$ . We prove that, for any $λ \in R ∖ {- 1, 0, 1}$ , \[ \max \{|A+_G A|, |A+_G λA|, |A\cdot_G A|\} \gg |G|^{6/11}. \]

math.CO2020

Arcs in $F_{q}^{2}$

Oliver Roche-Newton, Audie Warren

An arc is a subset of $F_{q}^{2}$ which does not contain any collinear triples. Let $A (q, k)$ denote the number of arcs in $F_{q}^{2}$ with cardinality $k$ . This paper is pr…

math.CO2019★ 2 cited

An Energy Bound in the Affine Group

Giorgis Petridis, Oliver Roche-Newton, Misha Rudnev +1

We prove a nontrivial energy bound for a finite set of affine transformations over a general field and discuss a number of implications. These include new bounds on growth in the a…

math.CO2019

New Expander Bounds from Affine Group Energy

Oliver Roche-Newton, Audie Warren

The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications o…

math.CO2019★ 1 cited

Four-term progression free sets with three-term progressions in all large subsets

Cosmin Pohoata, Oliver Roche-Newton

This paper is mainly concerned with sets which do not contain four-term arithmetic progressions, but are still very rich in three term arithmetic progressions, in the sense that al…

math.CO2018

Constructions for the Elekes-Szabó and Elekes-Rónyai problems

Mehdi Makhul, Oliver Roche-Newton, Audie Warren +1

We give a construction of a non-degenerate polynomial $F \in R [x, y, z]$ and a set $A$ of cardinality $n$ such that $\left|Z(F)\cap (A \times A \times A) \right| \gg n^{\frac{…

math.CO2018

An improved bound for the size of the set $A / A + A$

Oliver Roche-Newton

It is established that for any finite set of positive real numbers $A$ , we have $∣ A / A + A ∣ ≫ \frac{∣ A ∣ ^{\frac{3}{2} + \frac{1}{26}}}{lo g ^{1/2} ∣ A ∣} .$

math.CO2018

If $A + A$ is small then $AAA$ is superquadratic

Oliver Roche-Newton, Ilya D. Shkredov

This note proves that there exists positive constants $c_{1}$ and $c_{2}$ such that for all finite $A \subset R$ with $∣ A + A ∣ \leq ∣ A ∣^{1 + c_{1}}$ we have $∣ AAA ∣ ≫ ∣ A ∣^{2 + c_{2}}$ .

math.CO2018

Improved Bounds for Pencils of Lines

Oliver Roche-Newton, Audie Warren

We consider a question raised by Rudnev: given four pencils of $n$ concurrent lines in $R^{2}$ , with the four centres of the pencils non-collinear, what is the maximum possib…

math.CO2018★ 4 cited

On the size of the set $AA + A$

Oliver Roche-Newton, Imre Z. Ruzsa, Chun-Yen Shen +1

It is established that there exists an absolute constant $c > 0$ such that for any finite set $A$ of positive real numbers $∣ AA + A ∣ ≫ ∣ A ∣^{\frac{3}{2} + c} .$ On the other hand, we g…

math.CO2017

Variations on the sum-product problem II

Brendan Murphy, Oliver Roche-Newton, Ilya Shkredov

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the method…

math.CO2017★ 52 cited

New results on sum-product type growth over fields

Brendan Murphy, Giorgis Petridis, Oliver Roche-Newton +2

We prove a range of new sum-product type growth estimates over a general field $F$ , in particular the special case $F = F_{p}$ . They are unified by the theme…

Growth beyond exponent 3/2 for convexity and iterated sum sets

Geometric Sidon Problems

The minimum degree question for the Maker Breaker Domination Game

Large Convex sets in Difference sets

Local Differences Determined by Convex sets

A better than 3/2 exponent for iterated sums and products over R

Counting arcs in Fq2​

A convex set with a rich difference

Convexity, Squeezing, and the Elekes-Szabó Theorem

Convexity, Superquadratic Growth, and Dot Products

Additive and multiplicative Sidon sets

The Elekes-Szabó Problem and the Uniformity Conjecture

Sums, products and dilates on sparse graphs

Arcs in Fq2​

An Energy Bound in the Affine Group

New Expander Bounds from Affine Group Energy

Four-term progression free sets with three-term progressions in all large subsets

Constructions for the Elekes-Szabó and Elekes-Rónyai problems

An improved bound for the size of the set A/A+A

If A+A is small then AAA is superquadratic

Improved Bounds for Pencils of Lines

On the size of the set AA+A

Variations on the sum-product problem II

New results on sum-product type growth over fields

Growth beyond exponent $3/2$ for convexity and iterated sum sets

A better than $3/2$ exponent for iterated sums and products over $R$

Counting arcs in $F_{q}^{2}$

Arcs in $F_{q}^{2}$

An improved bound for the size of the set $A / A + A$

If $A + A$ is small then $AAA$ is superquadratic

On the size of the set $AA + A$