Structure vs randomness for bilinear maps, Discrete Analysis 2022:12, 21 pp. A _tensor_ can be thought of as a higher-dimensional analogue of a matrix (where a matrix is 2-dimensional). Tensors arise very naturally in the context of multilinear forms, since if $V_1,\dots,V_d$ are vector spaces over a field $\mathbb F$ with given bases, then a $d$-linear map $\mu$ from $V_1\times\dots\times V_d$ to $\mathbb F$ is given by a formula of the kind $\mu:(x_1,\dots,x_d)=\sum_{i_1,\dots,i_d}a_{i_1,\dots,i_d}x_{1,i_1}\dots x_{d,i_d}$, where $x_{i,j}$ is the $j$th coordinate of $x_i$. For matrices and bilinear forms, the notion of rank plays such a fundamental role that an obvious question is how to generalize it to tensors and multilinear forms. However, it turns out that there are several natural generalizations, and different notions of tensor rank are useful in different situations. It is therefore of great interest to try to understand how these notions relate to each other. One way to describe the rank of a bilinear form $\beta:V\times W\to\mathbb F$ is that it is the smallest $r$ such that we can write $\beta(v,w)$ as $\sum_{i=1}^ra_i(v)b_i(w)$ for some linear forms $a:V\to\mathbb F$ and $b:W\to\mathbb F$. Equivalently, we say that a bilinear form has rank 1 if it breaks up as a product of linear forms, and then the rank of a bilinear form is the smallest number of rank-1 forms needed to contain it in their linear span. If we try to generalize this to trilinear forms, we find that there are two ways that we might wish to define a rank-1 form. The more obvious, which leads to the notion of _tensor rank_ is to define it as a form $(u,v,w)\mapsto a(u)b(v)c(w)$, where $a$, $b$ and $c$ are linear forms. Another, which leads to the notion of _slice rank_, which played a central role in the developments surrounding the famous solution of the cap-set problem, is to define a rank-1 form as any trilinear form that is a product of a linear form in one variable and a bilinear form in the other two variables -- and is thus a product of two simpler forms. Clearly as $d$ increases, the possible ways of splitting up a $d$-linear form as a product of simpler multilinear forms increases as well, leading to several further notions of rank, of which we single out just one: a $d$-linear form is said to have _partition rank_ 1 if it can be written as a product of two multilinear forms in disjoint non-empty subsets of the variables. For instance, if $\beta$ is a bilinear form and $\tau$ a trilinear form, then the form $(v_1,v_2,v_3,v_4,v_5)\mapsto\beta(x_1,x_4)\tau(x_2,x_3,x_5)$ is a 5-linear form of partition rank 1. Another definition of rank for trilinear forms comes from the following observation in the bilinear case: if $\beta:V\times W\to\mathbb F_p$ is a bilinear form and $\omega_p=\exp(2\pi i/p)$, then the average $\mathbb E_{v\in V}\mathbb E_{w\in W}\omega_p^{\beta(v,w)}$ is equal to $p^{-\mathrm{rk}(\beta)}$. To see this, note that the expectation over $w$ is zero if $\beta(v,w)$ is not identically zero, and 1 otherwise. The quantity on the left-hand side generalizes very easily, leading Gowers and Wolf to define the _analytic rank_ of a $d$-linear form $\mu:V_1\times\dots\times V_d\mathbb F_p$ to be $-\log_p(\mathbb E_{v_1,\dots,v_d}\omega_p^{\mu(v_1,\dots,v_d)})$, and to prove that it has some good properties. (In particular, they proved that it is subadditive up to a constant factor. Later, [in an article published in this journal](https://discreteanalysisjournal.com/article/8654-the-analytic-rank-of-tensors-and-its-applications), Lovett removed the constant factor: that is, analytic rank is now known to be exactly subadditive.) The concept of analytic rank is closely related to the notion of the bias of a polynomial, which appeared in earlier work of Green, Tao, Kaufmann, Lovett, and others, which is a measure of how far from uniformly distributed the values taken by the polynomial are. As a result of that work, it became a central problem to determine how the partition rank and the analytic rank of a multilinear form are related. A straightforward argument (see Theorem 1.7 part (i) in the paper of Lovett mentioned above, which was also obtained independently by Kazhdan and Ziegler) shows that the analytic rank is at most the partition rank. A considerably less straightforward argument of Bhowmick and Lovett showed that in fact the two ranks are equivalent, in the sense that the partition rank is bounded as a function of the analytic rank. However, the bound they obtained was of Ackermann type. A breakthrough obtained independently by Janzer and Milićević improved this to a polynomial dependence. ([Janzer's paper is also published in this journal](https://discreteanalysisjournal.com/article/12935-polynomial-bound-for-the-partition-rank-vs-the-analytic-rank-of-tensors.).) The authors of this paper have obtained a further breakthrough, showing that analytic and partition rank are equal up to a constant factor, provided the characteristic of the underlying field is sufficiently large. This paper itself is not about that result, but about a precursor to it that deals with the trilinear case and does not require the assumption about the field (other than that it should not be $\mathbb F_2$). In the case of trilinear forms, partition rank is the same as slice rank: the authors show that the slice rank of a trilinear form is at most 8.13 times its analytic rank, and draw several interesting consequences. Just as a matrix corresponds to a linear map, a 3-tensor corresponds to a bilinear map -- hence the word "bilinear" in the title rather than "trilinear". (If $\tau:U\times V\times W\to\mathbb F$ is a trilinear form, we can define a bilinear map $\beta:U\times V\to W^*$ by $\beta(u,v)(w)=\tau(u,v,w)$.) This is a convenient perspective because the authors have used a further notion of rank (introduced by Kopparty, Zuiddam and the second author) that plays a crucial intermediate role in their proof. They define the _geometric rank_ of a bilinear map $\beta:U\times V\to W$ to be the codimension of the algebraic variety $\{(u,v)\in U\times V:\beta(u,v)=0\}$. (The dimension of a variety can be defined as the length of the longest strictly increasing chain of non-zero irreducible varieties contained in it.) The complete result they prove in this paper is that the slice rank is at most three times the geometric rank, which is at most 8.13 times the analytic rank. Thus, the main result of the paper, as well as being an important result in its own right, is a convincing definition of the importance of the notion of geometric rank. <iframe width="420" height="236" src="https://www.youtube.com/embed/HaMuRzdkuFQ" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

Small doubling, atomic structure and $\ell$-divisible set families, Discrete Analysis 2022:11, 16 pp. The following fact is a well-known and simple illustration of the power of linear algebra in solving combinatorial problems. Let $\mathcal A$ be a collection of subsets of $\{1,2,\dots,n\}$ such that any two subsets in the collection have an intersection of even size. Then $|\mathcal A|\leq 2^{\lfloor n/2\rfloor}$, with equality attained if $\mathcal A$ consists of all sets $A$ that are unions of the sets $\{1,2\}, \{3,4\}, \dots$, where the last set in the sequence is $\{n-1,n\}$ if $n$ is even, and $\{n-2,n-1\}$ if $n$ is odd. This can be proved by associating with each set $A$ its characteristic function, and regarding each function as an element of $\mathbb F_2^n$. If $x$ and $y$ are elements of $\mathbb F_2^n$, then we can define their scalar product $\langle x,y\rangle$ to be $\sum_{i=1}^nx_iy_i$, and the question we started with is then equivalent to asking how many vectors it is possible to choose if all scalar products are zero. If $\mathcal A\subset\mathbb F_2^n$ and we choose a maximal linearly independent set $x^{1},\dots,x^{m}$ of $\mathcal A$, then every element $x$ of $\mathcal A$ satisfies $m$ linearly independent conditions $\langle x^i,x\rangle=0$, so $\mathcal A$ lives inside a subspace of codimension at least $m$ and therefore dimension at most $n-m$. It follows that $m\leq n-m$ and therefore that $m\leq\lfloor n/2\rfloor$, which gives the required bound on $|\mathcal A|$. An obvious follow-up question is to ask what happens if the size of every intersection is a multiple of 3, or more generally of $\ell$ for some positive integer $\ell$. The obvious generalization of the construction above gives a lower bound of $2^{\lfloor n/\ell\rfloor}$, but the simple argument for the upper bound no longer works, and indeed in 1983 Frankl and Odlyzko constructed families that are considerably larger. They also conjectured that the lower bound holds if one strengthens the hypothesis by assuming that larger intersections have sizes divisible by $\ell$. That is, they conjectured that for every $\ell$ there exists $k$ such that if $\mathcal A$ is a collection of subsets of $\{1,2,\dots,n\}$ such that any $k$ sets in $\mathcal A$ have an intersection of cardinality divisible by $\ell$, then $|\mathcal A|\leq 2^{\lfloor n/\ell\rfloor}$. This paper proves the conjecture of Frankl and Odlyzko. Along the way, the authors prove a lemma of independent interest, the statement of which has a similar flavour to that of Freiman's theorem -- hence the phrase "small doubling" in the title of the paper. Given two vectors $x,y\in\{0,1\}^n$ write $x.y$ for their pointwise product, which corresponds to the intersection of the corresponding subsets of $\{1,2,\dots,n\}$. Also, given a subset $X\subset\{0,1\}^n$, write $\dim(X)$ for the dimension of the subspace spanned by $X$. The lemma concerns what can be said about $X$ if $\dim(X.X)$ is not much larger than $\dim(X)$ -- more precisely, the assumption is that $\dim(X.X)-\dim(X)\leq h$ for some constant $h$. A simple example of such a set is to take disjoint subsets $A_1,\dots,A_r$ of $\{1,2,\dots,n\}$ and to take the space $V$ of all vectors that are constant on each set $A_i$, and to add to it a subspace $W$ of dimension $\sqrt h$ of vectors supported outside $A_1\cup\dots\cup A_r$. The space $V+W$ has dimension $r+\sqrt h$, and the space $(V+W)(V+W)=V.V+W.W=V+W.W$ has dimension $r+\dim(W.W)$, which is at most $r+h$, since if $w_1,\dots,w_t$ is a basis of $W$, then the vectors $w_i.w_j$ are a basis of $W.W$. The lemma proved by the authors is that, at least qualitatively speaking, all examples are of this kind. They show that if $\dim(X.X)-\dim(X)\leq h$, then there exist disjoint subsets $A_1,\dots,A_r\subset\{1,2,\dots,n\}$ with $r=\dim(X)$ such that every vector in $X$ is constant on each $A_i$, and the restrictions of the $x\in X$ to the complement of $A_1\cup\dots\cup A_r$ span a subspace of dimension at most $2h$. <iframe width="420" height="237" src="https://www.youtube.com/embed/bSkRwZcOJac" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

Cycle type of random permutations: a toolkit, Discrete Analysis 2022:9, 36 pp. What does a random permutation look like? This general question has led to a great deal of research, and this article provides a convenient single reference for many results, some new, others well known but with more unified proofs. Throughout the paper, the author is motivated by an analogy between how the cycle type of a random permutation is distributed, and how the sizes of prime factors of a random integer (between $n$ and $2n$ for large $n$, say) is distributed. Although the methods do not carry over from one domain to the other, the number-theoretic results turn out to be a surprisingly good guide to what will be true in the case of permutations. Some of the questions tackled are the following. Given a positive integer $j$, how is the number of cycles of length $j$ distributed? Given a set $I$, how is the number of cycles with length in $I$ distributed? If $I_1,\dots,I_k$ are disjoint sets, then what does the joint distribution of the numbers of cycles with lengths in $I_j$ look like? And what is the effect of conditioning on the total number of cycles. Let $C_j(\sigma)$ be the number of cycles of $\sigma$ of length $j$. For small $j$ heuristic arguments lead quickly to the conclusion that $C_j(\sigma)$ should be approximately Poisson with mean $1/j$. (To see that this should be the mean, note that the probability that 1 belongs to a $j$-cycle of $\sigma$ is roughly the probability that $\sigma^j(1)=1$, which is roughly $1/n$. So the expected number of members of $j$-cycles is roughly 1, and therefore the expected number of $j$-cycles is roughly $1/j$.) It also leads to the conclusion that if $j_1,\dots,j_k$ are small distinct integers, then $C_{j_1}(\sigma),\dots,C_{j_k}(\sigma)$ should be approximately independent. From this it follows that for small sets $I$ of small numbers, $C_I(\sigma)$, the number of cycles with lengths in $I$, ought to be approximately Poisson with mean $\sum_{j\in I}1/j$. These heuristics are referred to as the _Poisson model_, and much of the paper is devoted to an analysis of when the Poisson model gives good results and when it needs refining, as it does when long cycles are involved. Many of the results that are scattered through the literature are proved using generating functions. By contrast, the methods here are mainly direct probabilistic and combinatorial ones, which give uniform and explicit results. The outcome represents the state of the art for elementary methods and provides an excellent benchmark that any more sophisticated method should be expected to beat. The paper will be useful both for bringing the results together under one roof and also for introducing a general and flexible approach to solving them.

We establish a sharp sufficient condition for groups acting on trees to be highly transitive when the action on the tree is minimal of general type. This gives new examples of highly transitive groups, including icc non-solvable Baumslag-Solitar groups, thus answering a question of Hull and Osin.

Decidability of the isomorphism and the factorization between minimal substitution subshifts, Discrete Analysis 2022:7, 65 pp. Symbolic dynamics is the study of topological dynamical systems $(X,S)$ where $X$ is a shift-invariant space of singly or doubly infinite sequences of words in a finite alphabet, with the product topology, and $S$ is the shift map. Such systems are called _subshifts_. Subshifts are important partly because many dynamical systems can be represented in this form (very roughly, given a map $T$ defined on some manifold $M$, one divides $M$ up into a finite number of small regions and associates with each point of the manifold the sequence of regions it visits under iterates of $T$), and partly because they are interesting in their own right and lead to many attractive questions. A very basic question is to classify subshifts. However, that turns out to be hopeless, as the problem of determining whether two subshifts are isomorphic has been shown to be undecidable. In the light of this, attention has turned to investigating more specific classes of subshifts. Arguably the main open problem of this type is to determine whether the problem is decidable for a class of subshifts that are created as follows. Given an alphabet $\Sigma$ and a 01-matrix $A$ indexed by $\Sigma\times\Sigma$, one can obtain a subshift by taking all sequences $(x_n)$ of elements of $\Sigma$ such that $A_{x_{n-1}x_n}=1$ for every $n$. Such subshifts are called subshifts _of finite type_. Since subshifts of finite type are defined by a finite amount of data, it makes sense to ask whether isomorphism of such subshifts (together with an irreducibility condition) is decidable. This paper looks at another important and intensively studied class of subshifts, known as substitution subshifts, which are obtained as follows. One begins with an alphabet $A$, a symbol $a\in A$, and a set of substitution rules that provide, for each element of $A$, a finite sequence of elements that should replace it. If we start with just a singleton sequence $a$, and if the sequence that replaces $a$ begins with $a$, then these rules generate an infinite sequence. For example, the well-known Morse sequence 01101001100101101001... is generated by the initial element 0 and the rules that 0 is replaced by 01 and 1 is replaced by 10. To turn this into a subshift, one takes $X$ to be the space of all sequences that belong to the closure of the set of shifts of the sequence $x$ we have generated. This consists of all sequences $y$ such that every restriction of $y$ to an interval of $y$ agrees with some restriction of $x$ to an interval (necessarily of the same length). Again, only a finite amount of data is needed to define a substitution subshift, so again it makes sense to ask if there is an algorithm to determine whether two substitution subshifts are isomorphic. This problem is not quite as widely stated as the problem for irreducibile subshifts of finite type, but substitution subshifts are a fundamental source of examples of abstract dynamical systems, making this a very important question. The paper solves the question for a wide class of substitution subshifts, known as minimal substitution subshifts. In fact, it does considerably more than this: given two such subshifts, it shows that one can compute all the morphisms between them. The paper also gives bounds on the sizes of these morphisms, and hence on the complexity of the algorithms needed to compute them. (These bounds are quite large but not ridiculously so -- they are given by composing a bounded number of exponentials.) Several questions are left open, but the main results of this paper are a big step forward in the area, building on a great deal of earlier work by the authors and others. They are likely to be of interest to researchers in dynamics, cellular automata, computability, and complexity theory, as well as solving questions that are natural and interesting in their own right.

Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Tur\'an numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $\lambda_k \ge k^{-2k^2}$ such that for any $k$-complex $\mathcal{S}$, every $k$-complex on $n \ge n_0(\mathcal{S})$ vertices with at least $n^{k+1 - \lambda_k}$ facets contains a homeomorphic copy of $\mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $\lambda_1$ is a result of Mader from 1967, and the existence of $\lambda_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, \dots, V_r$ is said to be $d$-trace-bounded if for each $2 \le i \le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 \cup V_2 \cup \dots \cup V_i$. Our second main result is the following estimate for the Tur\'an numbers of degenerate trace-bounded hypergraphs: for all $r \ge 2$ and $d\in\mathbb N$, there is an $\alpha_{r,d} \ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n \ge n_0(H)$ vertices with at least $n^{r - \alpha_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).

Random cones in high dimensions I: Donoho-Tanner and Cover-Efron cones, Discrete Analysis 2020:5, 44 pp. As its title makes clear, this paper is about random high-dimensional cones. A _cone_ in $\mathbb R^d$ is a subset that is closed under addition and under multiplication by non-negative scalars. To define a random cone is less easy, since there is no single way of defining a probability measure on the set of all cones that stands out as being the most natural. Such a situation can be regarded as an opportunity: one can try to find definitions that lead to interesting questions. One obvious randomized method of obtaining a cone in $\mathbb R^d$ is to pick $n$ vectors $v_1,\dots,v_n$ independently at random from some distribution on $\mathbb R^d$ and to take the cone that they generate. And an obvious distribution to take is the standard Gaussian distribution. There is also the question of which $n$ to take. If $n$ is smaller than $d$, then the cone will not be of full dimension, and if $n$ is significantly larger than $d$, then with high probability it will be the whole of $\mathbb R^d$. More precisely, a theorem of Schläfi from the 19th century gives that $n$ hyperplanes through the origin but otherwise in general position divide $\mathbb R^d$ into $C(n,d)=2\sum_{i=0}^{d-1}\binom{n-1}i$ regions. If the normal vectors are $v_1,\dots,v_n$, then each region is of the form $\{x:\forall i\ \langle x,\epsilon_iv_i\rangle>0\}$ for some choice of signs $\epsilon_1,\dots,\epsilon_n$. It follows that there are precisely $C(n,d)$ choices of signs $\epsilon_1,\dots,\epsilon_n$ for which there exists $x$ such that $\langle x,\epsilon_iv_i\rangle>0$. It follows that if $v_1,\dots,v_n$ are any $n$ vectors in general position and $\epsilon_1,\dots,\epsilon_n$ are random signs, then the probability that there is a hyperplane such that all the vectors $\epsilon_iv_i$ lie on one side of the hyperplane is $C(n,d)/2^n$. And from that it follows that if $v_1,\dots,v_n$ are chosen independently at random from a distribution that is centrally symmetric and absolutely continuous with respect to Lebesgue measure, then the probability that they lie to one side of a hyperplane, and therefore generated a non-trivial cone, is also $C(n,d)/2^n$. This implies that there is an important change when $n$ passes $2d$. If $d=\delta n$ for some $\delta<1/2$. then $C(n,d)/2^n$ is exponentially small, so the cone generated by $v_1,\dots,v_n$ is all of $\mathbb R^d$ with extremely high probability, whereas if $\delta>1/2$ then $1-C(n,d)/2^n$ is exponentially small, so the cone is almost certainly proper. This threshold has been shown to be a phase transition for a number of important parameters associated with random cones. What is achieved in this paper is a set of much more precise results concerning how the changes take place over the "critical window" -- that is, how the parameters depend on $c$ if $n=2d+c\sqrt d+o(\sqrt d)$. The Donoho-Tanner random cones in the title are the ones just described. The Cover-Efron cones are ones where one conditions on the cone not being all of $\mathbb R^d$. The authors prove results for both models. One of the parameters they investigated is the number of faces of dimension $k$ for small $k$. This was known to be around $\binom nk$ if $\delta>1/2$ and around $(2\delta)^k\binom nk$ if $\delta<1/2$, but the authors provide a formula for the dependence on $c$ in the critical window (which in the case of the Donoho-Tanner random cones turns out to be given by the cumulative distribution function of a normal distribution). They also look at _quermassintegrals_: the $k$th quermassintegral of a random cone is the probability that it intersects non-trivially with a random subspace of codimension $k$. Previous work in this area has tended to rely on detailed estimates for sums of binomial coefficients (as the simple argument given earlier exemplifies). The main idea that enables the authors to go beyond this is to interpret the quantities they are looking at in terms of binomial random variables, which allows them to make use several less elementary probabilistic tools.

We determine the asymptotic behavior of twisted traces of singular moduli with a power-saving error term in both the discriminant and the order of the pole at $i\infty$. Using this asymptotic formula, we obtain an exact formula for these traces involving the class number and a finite sum involving the exponential function evaluated at CM points.