H. Helfgott

activity

19982025

most citedMinor arcs for Goldbach's problem

50 citations · 353 across the 37 of their papers we have counts for

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13 papers · 1 filter

math.GR2023

New dimensional estimates for subvarieties of linear algebraic groups

Jitendra Bajpai, Daniele Dona, Harald Andrés Helfgott

For every connected, almost simple linear algebraic group $G \leq GL_{n}$ over a large enough field $K$ , every subvariety $V \subseteq G$ , and every finite generating set $A\…

math.GR2021★ 2 cited

Growth estimates and diameter bounds for untwisted classical groups

Jitendra Bajpai, Daniele Dona, Harald Andrés Helfgott

Babai's conjecture states that, for any finite simple non-abelian group $G$ , the diameter of $G$ is bounded by $(lo g ∣ G ∣)^{C}$ for some absolute constant $C$ . We prove that, for an…

math.GR2019

Growth and expansion in algebraic groups over finite fields

Harald Andres Helfgott

This is a brief introduction to the study of growth in groups of Lie type, with $S L_{2} (F_{q})$ and some of its subgroups as the key examples. They are an edited version of th…

math.GR2018

Growth in linear algebraic groups and permutation groups: towards a unified perspective

Harald A. Helfgott

By now, we have a product theorem in every finite simple group $G$ of Lie type, with the strength of the bound depending only in the rank of $G$ . Such theorems have numerous conseq…

math.GR2017★ 21 cited

Graph isomorphisms in quasi-polynomial time

Harald Andrés Helfgott, Jitendra Bajpai, Daniele Dona

Let us be given two graphs $Γ_{1}$ , $Γ_{2}$ of $n$ vertices. Are they isomorphic? If they are, the set of isomorphisms from $Γ_{1}$ to $Γ_{2}$ can be identified with a coset $H \cdot π$ insi…

math.GR2015

Soficity, short cycles and the Higman group

Harald A. Helfgott, Kate Juschenko

This is a paper with two aims. First, we show that the map from $Z / p Z$ to itself defined by exponentiation $x \to m^{x}$ has few $3$ -cycles -- that is to say, the nu…

math.GR2013

Random generators of the symmetric group: diameter, mixing time and spectral gap

Harald A. Helfgott, Ákos Seress, Andrzej Zuk

Let $g$ , $h$ be a random pair of generators of $G = S y m (n)$ or $G = A l t (n)$ . We show that, with probability tending to $1$ as $n \to \infty$ , (a) the diameter of $G$ with respect to $S…

math.GR2013★ 2 cited

Growth in groups: ideas and perspectives

H. A. Helfgott

This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics a…

math.GR2011★ 8 cited

On the diameter of permutation groups

Harald A. Helfgott, Akos Seress

Given a finite group $G$ and a set $A$ of generators, the diameter diam $(Γ (G, A))$ of the Cayley graph $Γ (G, A)$ is the smallest $ℓ$ such that every element of $G$ can be expresse…

math.GR2010★ 24 cited

Growth in solvable subgroups of GL_r(Z/pZ)

Nick Gill, Harald Andres Helfgott

Let $K = Z / pZ$ and let $A$ be a subset of $\GL_r(K)$ such that $< A >$ is solvable. We reduce the study of the growth of $A$ under the group operation to the nilpotent setting. Specifi…

math.GR2010

Growth of small generating sets in SL_n(Z/pZ)

Nick Gill, Harald Andres Helfgott

Let G=SL_n. Let K=Z/pZ, p a prime. Let A\subset G(K) generate G(K). Suppose that |A|<p^{n+1-δ}, delta>0. Then |A A A|>>|A|^{1+ε}, where epsilon>0 and the implied constant depend on…

math.GR2008★ 14 cited

Growth in SL_3(Z/pZ)

H. A. Helfgott

Let G=SL_3(Z/pZ), p a prime. Let A be a set of generators of G. Then A grows under the group operation. To be precise: denote by |S| the number of elements of a finite set S. Assum…

math.GR2005★ 44 cited

Growth and generation in SL_2(Z/pZ)

H. A. Helfgott

We show that every subset of SL_2(Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL_2(Z/pZ), every ele…