Sean K. Eberhard

Queen's University Belfast | QUB · School of Mathematics and Physics

Let $G$ be a finite classical group generated by transvections, i.e., one of $\operatorname{SL}_n(q)$, $\operatorname{SU}_n(q)$, $\operatorname{Sp}_{2n}(q)$, or $\operatorname{O}^\pm_{2n}(q)$ ($q$ even), and let $X$ be a generating set for $G$ containing at least one transvection. Building on work of Garonzi, Halasi, and Somlai, we prove that the d...

A transversal in an latin square is a collection of entries not repeating any row, column, or symbol. Kwan showed that almost every latin square has transversals as . Using a loose variant of the circle method we sharpen this to . Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin...

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essential...

For G a finite group, let $$d_2(G)$$ d 2 ( G ) denote the proportion of triples $$(x, y, z) \in G^3$$ ( x , y , z ) ∈ G 3 such that $$[x, y, z] = 1$$ [ x , y , z ] = 1 . We determine the structure of finite groups G such that $$d_2(G)$$ d 2 ( G ) is bounded away from zero: if $$d_2(G) \ge \epsilon > 0$$ d 2 ( G ) ≥ ϵ > 0 , G has a class-4 nilpotent...

A transversal in an $n \times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n \times n$ latin square has $\bigl((1 + o(1)) n^2 / e\bigr)^n$ transversals as $n \to \infty$. Using a loose variant of the circle method we sharpen this to $(e^{-1/2} + o(1)) n!^2 / n^n$. Our method...

Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle me...

This paper is a follow-up to (arXiv:2203.03687), in which the first author studied primitive association schemes lying between a tensor power $\mathcal{T}_m^d$ of the trivial association scheme and the Hamming scheme $\mathcal{H}(m,d)$. A question which arose naturally in that study was whether all primitive fusions of $\mathcal{T}_m^d$ lie between...

Suppose $\pi$ and $\pi'$ are two random elements of $S_n$ with constrained cycle types such that $\pi$ has $x n^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise $\pi'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show that the events that $G = \langle \pi, \pi' \rangle$ is transitive and $G \geq A_n$ both have probability approxi...

Form an n × n matrix by drawing entries independently from {±1} (or another fixed nontrivial finitely supported distribution in Z) and let φ be the characteristic polynomial. We show, conditionally on the extended Riemann hypothesis, that with high probability φ is irreducible and Gal(φ) ≥ An.

We describe primitive association schemes $\mathfrak{X}$ of degree $n$ such that $\mathrm{Aut}(\mathfrak{X})$ is imprimitive and $|\mathrm{Aut}(\mathfrak{X})| \geq \exp(n^{1/8})$, contradicting a conjecture of Babai. This and other examples we give are the first known examples of nonschurian primitive coherent configurations (PCC) with more than a...

Let $$G = {\text {SCl}}_n(q)$$ G = SCl n ( q ) be a quasisimple classical group with n large, and let $$x_1, \ldots , x_k \in G$$ x 1 , … , x k ∈ G be random, where $$k \ge q^C$$ k ≥ q C . We show that the diameter of the resulting Cayley graph is bounded by $$q^2 n^{O(1)}$$ q 2 n O ( 1 ) with probability $$1 - o(1)$$ 1 - o ( 1 ) . In the particula...

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq \epsilon > 0$, $G$ has a class-4 nilpotent normal subgroup $H$ such that $[G : H] $ and $|\gamma_4(H)|$ are both bounded in...

We adapt the theory of partition rank and analytic rank to the category of abelian groups. If $A_1, \dots, A_k$ are finite abelian groups and $\phi : A_1 \times \cdots \times A_k \to \mathbf{T}$ is a multilinear map, where $\mathbf{T} = \mathbf{R}/\mathbf{Z}$, the bias of $\phi$ is defined to be the average value of $\exp(i 2 \pi \phi)$. If the bia...

We prove a conjecture of Helfgott and Lindenstrauss on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of $\mathrm{GL}_n(\mathbf{F})$ for any field $\mathbf{F}$ such that $|A^3| \leq K|A|$. Then there are subgroups $H \trianglelefteq \Gamma \leq \langle A \rangle$ such that...

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essential...

Here is a simplified proof that every sharply transitive subset of PGL 2 ( K ) is a coset of a subgroup, for every finite field K .

Here is a simplified proof that every sharply transitive subset of $\mathrm{PGL}_2(K)$ is a coset of a subgroup.

A family of vectors in [ k ] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [ k ] n invariant under a transitive group of symmetries is o ( k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our mai...

Define (Xn) on Z/qZ by Xn+1=2Xn+bn, where the steps bn are chosen independently at random from -1,0,+1. The mixing time of this random walk is known to be at most 1.02log2q for almost all odd q (Chung, Diaconis, Graham in Ann Probab 15(3):1148–1165, 1987), and at least 1.004log2q (Hildebrand in Proc Am Math Soc 137(4):1479–1487, 2009). We identify...

Let ${\mathbb{P}}(ord\pi = ord\pi ')$ be the probability that two independent, uniformly random permutations of [ n ] have the same order. Answering a question of Thibault Godin, we prove that ${\mathbb{P}}(ord\pi = ord\pi ') = {n^{ - 2 + o(1)}}$ and that ${\mathbb{P}}(ord\pi = ord\pi ') \ge {1 \over 2}{n^{ - 2}}lg*n$ for infinitely many n . (Here...

Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on the extended Riemann hypothesis, with high probability $\phi$ is irreducible and $\mathrm{Gal}(\phi) \geq A_n$.

Let $G = \mathrm{SCl}_n(q)$ be a quasisimple classical group with $n$ large, and let $x_1, \dots, x_k \in G$ random, where $k \geq q^C$. We show that the diameter of the resulting Cayley graph is bounded by $q^2 n^{O(1)}$ with probability $1 - o(1)$. In the particular case $G = \mathrm{SL}_n(p)$ with $p$ a prime of bounded size, we show that the sa...

Define $(X_n)$ on $\mathbf{Z}/q\mathbf{Z}$ by $X_{n+1} = 2X_n + b_n$, where the steps $b_n$ are chosen independently at random from $-1, 0, +1$. The mixing time of this random walk is known to be at most $1.02 \log_2 q$ for almost all odd $q$ (Chung--Diaconis--Graham, 1987), and at least $1.004 \log_2 q$ (Hildebrand, 2008). We identify a constant $...

Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping if and only if its Sylow $2$-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circl...

It is well known that the proportion of pairs of elements of SL ⁡ ( n , q ) \operatorname {SL}(n,q) which generate the group tends to 1 1 as q n → ∞ q^n\to \infty . This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give a proof of this theorem which does not depend on the classification. An essential step i...

A family of vectors $A \subset [k]^n$ is said to be intersecting if any two elements of $A$ agree on at least one coordinate. We prove, for fixed $k \ge 3$, that the size of a symmetric intersecting subfamily of $[k]^n$ is $o(k^n)$, which is in stark contrast to the case of the Boolean hypercube (where $k =2$). Our main contribution addresses limit...

For any finite group $G$, and any positive integer $n$, we construct an association scheme which admits the diagonal group $D_n(G)$ as a group of automorphisms. The rank of the association scheme is the number of partitions of $n$ into at most $|G|$ parts, so is $p(n)$ if $|G|\ge n$; its parameters depend only on $n$ and $|G|$. For $n=2$, the assoc...

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $\pi, \pi' \in S_n$, we prove that $\pi$ and a random conjugate of $\pi'$ are likely to generate at least $A_n$ provided only that $\pi$ and $\pi'$ have not too many fixed points and not too many $2$-cycles. As an application, we investigate the follow...

It is well known that the proportion of pairs of elements of $\operatorname{SL}(n,q)$ which generate the group tends to $1$ as $q^n\to \infty$. This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give a proof of this theorem which does not depend on the classification. An essential step in our proof is an est...

The Ewens sampling formula with parameter $\alpha$ is the distribution on $S_n$ which gives each $\pi\in S_n$ weight proportional to $\alpha^{C(\pi)}$, where $C(\pi)$ is the number of cycles of $\pi$. We show that, for any fixed $\alpha$, two Ewens-random permutations generate at least $A_n$ with high probability. More generally we work out how man...

We consider the probability p(Sn ) that a pair of random permutations generates either the alternating group An or the symmetric group Sn . Dixon (1969) proved that p(Sn ) approaches 1 as n→∞ and conjectured that p(Sn ) = 1 − 1/n+o(1/n). This conjecture was verified by Babai (1989), using the Classification of Finite Simple Groups. We give an eleme...

Consider the Cayley graph of $S_n$ generated by a random pair of elements $x,y$. Conjecturally, the girth of this graph is $\Omega(n \log n)$ with probability tending to $1$ as $n\to\infty$. We show that it is at least $\Omega(n^{1/3})$.

We study additive properties of the set $S$ of bijections (or permutations) $\{1,\dots,n\}\to G$, thought of as a subset of $G^n$, where $G$ is an arbitrary abelian group of order $n$. Our main result is an asymptotic for the number of solutions to $\pi_1 + \pi_2 + \pi_3 = f$ with $\pi_1,\pi_2,\pi_3\in S$, where $f:\{1,\dots,n\}\to G$ is an arbitar...

Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the sums of the form $\sum_{x \in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the...

We give a brief exposition and proof of the arithmetic regularity lemma of Green and Tao in the abelian ($U^2$) case, over $\{1,\dots,N\}$. This may be useful to those who need just the $U^2$ case of the lemma, as the general case is significantly more involved. It may also be useful as an introduction to the general case. No originality is claimed...

In the first paper in this series we estimated the probability that a random permutation $\pi\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $\pi$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\dots,k_m$, where $k_1+\cdots+k_m=n$. We deduce an estimate for the...

We prove the following one-sided product-mixing theorem for the alternating group: Given subsets $X,Y,Z \subset A_n$ of densities $\alpha,\beta,\gamma$ satisfying $\min(\alpha\beta,\alpha\gamma,\beta\gamma)\gg n^{-1}(\log n)^7$, there are at least $ (1+o(1))\alpha\beta\gamma |A_n|^3$ solutions to $xy=z$ with $x\in X, y\in Y, z\in Z$. One consequenc...

Recently Breuillard and Tointon showed that one reasonable formulation of the polynomial Freiman-Ruzsa conjecture fails for nonabelian groups. We improve and simplify their construction.

We prove an asymptotic for the number of additive triples of bijections $\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs of bijections $\pi_1,\pi_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ such that the pointwise sum $\pi_1+\pi_2$ is also a bijection. This problem is equivalent to counting the number of orthomorphisms or...

We say that permutations $\pi_1,\dots, \pi_r \in \mathcal{S}_n$ invariably generate $\mathcal{S}_n$ if, no matter how one chooses conjugates $\pi'_1,\dots,\pi'_r$ of these permutations, $\pi'_1,\dots,\pi'_r$ generate $\mathcal{S}_n$. We show that if $\pi_1,\pi_2,\pi_3$ are chosen randomly from $\mathcal{S}_n$ then, almost surely as $n \rightarrow \...

Let $i(n,k)$ be the proportion of permutations $\pi \in \mathcal {S}_n$ having an invariant set of size $k$. In this note, we adapt arguments of the second author to prove that $i(n,k) \asymp k^{-\delta } (1+\log k)^{-3/2}$ uniformly for $1\leq k\leq n/2$, where $\delta = 1 - \frac {1 + \log \log 2}{\log 2}$. As an application, we show that the pro...

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let ${{\mathcal P}}\subset (0,1]$ be the set of commuting probabilities of all finite groups. We prove that every point of ${{\mathcal P}}$ is nearly an Egyptian fraction of bounded complexity. As a corollary, we deduce two...

The purpose of this paper is to establish a method of obtaining closed-form solutions in isotropic hyperelasticity using the complementary energy, the Legendre transform of the strain energy function. Using the complementary energy, the stress becomes the independent variable and the strain the dependent variable. Some of the implications of this f...

Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$ is a multiplicative F{\o}lner sequence in $\mathbf{N}$ then $F_m$ has no $k$-sum-free subset of size greater...

Answering a question of P. Erdos from 1965, we show that for every eps>0 there is a set A of n integers with the following property: every subset A' of A with at least (1/3 + eps)n elements contains three distinct elements x,y,z with x + y = z.

We generalise an argument of Leader, Russell, and Walters to show that almost all sets of d + 2 points on the (d - 1)-sphere S^{d-1} are not contained in a transitive set in some R^n.

The motion of a projectile in a uniform gravitational field loses its symmetry when a resisting force is present, essentially because Newton's law loses its reversibility. Allowing the magnitude of the resistance to depend arbitrarily on speed, the motion is governed by two coupled first-order non-linear differential equations. Though intractable t...