Let $\mathcal{L}$ be a family of lines and let $\mathcal{P}$ be a family of $k$-planes in $\mathbb{F}^n$ where $\mathbb{F}$ is a field. In our first result we show that the number of joints formed by a $k$-plane in $\mathcal{P}$ together with $(n-k)$ lines in $\mathcal{L}$ is $O_n(|\mathcal{L}||\mathcal{P}|^{1/(n-k)}$). This is the first sharp result for joints involving higher-dimensional affine subspaces, and it holds in the setting of arbitrary fields $\mathbb{F}$. In contrast, for our second result, we work in the three-dimensional Euclidean space $\mathbb{R}^3$, and we establish the Kakeya-type estimate \begin{equation*}\sum_{x \in J} \left(\sum_{\ell \in \mathcal{L}} \chi_\ell(x)\right)^{3/2} \lesssim |\mathcal{L}|^{3/2}\end{equation*} where $J$ is the set of joints formed by $\mathcal{L}$; such an estimate fails in the setting of arbitrary fields. This result strengthens the known estimates for joints, including those counting multiplicities. Additionally, our techniques yield significant structural information on quasi-extremisers for this inequality.

We say that a family $\\mathcal F$ of $k$-element sets is a {\\it $j$-junta} if\nthere is a set $J$ of size $j$ such that, for any $F$, its presence in\n$\\mathcal F$ depends on its intersection with $J$ only. Approximating arbitrary\nfamilies by $j$-juntas with small $j$ is a recent powerful technique in\nextremal set theory.\n The weak point of all known junta approximation results is that they work in\nthe range $n>Ck$, where $C$ is an extremely fast growing function of the input\nparameters, such as the quality of approximation or the number of families we\nsimultaneously approximate.\n We say that a family $\\mathcal F$ is {\\it shifted} if for any\n$F=\\{x_1,\\ldots, x_k\\}\\in \\mathcal F$ and any $G =\\{y_1,\\ldots, y_k\\}$ such\nthat $y_i\\le x_i$, we have $G\\in \\mathcal F$. For many extremal set theory\nproblems, including the Erd\\H os Matching Conjecture, or the Complete\n$t$-Intersection Theorem, it is sufficient to deal with shifted families only.\n In this paper, we present very general approximation by juntas results for\nshifted families with explicit (and essentially linear) dependency on the input\nparameters. The results are best possible up to some constant factors.\nMoreover, they give meaningful statements for almost all range of values of\n$n$. The proofs are shorter than the proofs of the previous approximation by\njuntas results and are completely self-contained.\n As an application of our junta approximation, we give a nearly-linear bound\nfor the multi-family version of the Erd\\H os Matching Conjecture. More\nprecisely, we prove the following result. Let $n\\ge 12sk\\log(e^2s)$ and suppose\nthat the families $\\mathcal F_1,\\ldots, \\mathcal F_s\\subset {[n]\\choose k}$ do\nnot contain $F_1\\in\\mathcal F_1,\\ldots, F_s\\in \\mathcal F_s$ such that $F_i$'s\nare pairwise disjoint. Then $\\min_{i}|\\mathcal F_i|\\le {n\\choose\nk}-{n-s+1\\choose k}.$\n

Sylvester-Gallai type theorems for quadratic polynomials, Discrete Analysis 2020:13, 34 pp. The [Sylvester-Gallai theorem](https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem) asserts that given $n$ points in a plane that are not contained in a line, then there must be some line that contains exactly two of the points. This result is one of the basic results of combinatorial geometry, with several proofs, and it has been strengthened and generalized in a number of different ways -- see for example [a paper published recently in this journal](https://discreteanalysisjournal.com/article/11949-on-sets-defining-few-ordinary-hyperplanes). If we dualize the original theorem, we arrive at the statement that if $n$ lines in the plane do not all meet at a point, then there must be a point that is contained in exactly two of the lines. If we think of the lines as graphs of polynomials of degree 1, then we can ask whether there are higher-degree generalizations of the theorem. That is the theme of this paper, which proves a quadratic analogue of the theorem. To understand the statement that the author proves, it is helpful to reformulate a little further the linear version. Observe first that the original, non-dualized version of the theorem works just as well if the points live in an arbitrary vector space. Indeed, given $n$ points in a space $V$, we can find a projection to a 2-dimensional subspace of $V$ such that the images of any three non-collinear points are non-collinear, and hence such that the collinearity relations are all the same. Therefore, if no line contains exactly two points in the original set, then no line contains exactly two points in the projected set, from which it follows that the points in the projected set all lie in a line, from which it follows (since we did not create any new collinear triples) that the points in the original set all lie in a line. The above statement is an affine statement, but we can replace it by an equivalent linear statement, which is that if $x_1,\dots,x_n$ are non-zero vectors in a (real) vector space $V$ such that for any two of the vectors $x_i,x_j$ there is a third, $x_k$ that lies in their span, then there is a 2-dimensional subspace that contains all the vectors. (To see the equivalence, project the vectors from zero to a random affine subspace, and then belonging to a 2-dimensional subspace translates into collinearity.) From this we can deduce that if $\mathcal L$ is a finite collection of linear polynomials in $n$ variables $x_1,\dots,x_n$, and if for any two polynomials $L_1,L_2$ from the collection there is a third polynomial $L_3$ from the collection such that $L_3(x_1,\dots,x_n)=0$ whenever $L_1(x_1,\dots,x_n)=L_2(x_1,\dots,x_n)=0$, then $\mathcal L$ is contained in a 2-dimensional subspace. Indeed, if the coefficients of $L_3$ are a linear combination of those of $L_1$ and $L_2$, then any solution of $L_1$ and $L_2$ is clearly a solution of $L_3$, and since the polynomials are linear, the converse is true as well. The author generalizes this statement by replacing $\mathcal L$ with a finite collection $\mathcal Q$ of _quadratic_ polynomials in $n$ variables (in fact over $\mathbb C$) which are assumed to be irreducible. The conclusion is now that the subspace generated by $\mathcal Q$ is of bounded dimension. It is always interesting to prove non-trivial higher-degree generalizations of linear results, but that is far from the only, or even main, motivation for this paper, which comes from computational complexity. An important theme in that field is _polynomial identity testing_ -- you are given a formula for a multivariable polynomial, and you would like to determine (with an efficient algorithm) whether that polynomial is identically zero. (This is of course equivalent to the problem of testing whether two polynomials agree as functions.) More precisely, you are given an _arithmetic circuit_, which is a directed graph with $n$ input nodes and one output node, with each non-input node labelled with $+$ or $\times$). Another way of thinking about it is that you have a sequence of polynomials, starting with $x_1,\dots,x_n$, with each subsequent one obtained by adding or multiplying some of the earlier ones together. It turns out that for certain special classes of arithmetic circuits, if they compute the zero polynomial, then one can obtain Sylvester-Gallai-type conditions concerning linear forms that are used to build it up. Applying the Sylvester-Gallai theorem one deduces that these linear forms live in a low-dimensional space, which allows one to apply an invertible linear transformation and obtain a polynomial in a bounded number of variables, for which identity testing is easy. The circuits in question have depth 3 -- this means that the longest directed path in the directed graph has at most three edges -- as well as satisfying certain additional conditions. Significant additional difficulties arise if one increases the depth to 4, but there is an ambitious conjecture of Gupta, which is thought to be hard, that attempts to resolve them. He also formulated a sequence of related conjectures in the direction of his main conjecture, which are generalizations of the Sylvester-Gallai theorem. One of them is that the (dualized) Sylvester-Gallai theorem holds for polynomials of bounded degree. By proving the result for quadratic polynomials, this paper establishes the first non-trivial case of the conjecture. The proof of the theorem is also very interesting, in that it brings tools from theoretical computer science to bear on a problem in combinatorial (and algebraic) geometry. Particularly striking is that in order to prove the result, the author makes significant use of a "robust" version of the linear theorem. In the original formulation it states that if we have a set $S$ of $n$ points for which we assume only that for each point $x$ in $S$ there are at least $\delta n$ other points $y$ in $S$ such that the line through $x$ and $y$ passes through at least one further point in $S$, then $S$ is contained in an $O(1/\delta)$-dimensional space. It seems that extending the results to higher-degree polynomials would require significant further ideas, but this is an important first step.

The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, Discrete Analysis 2020:5, 42 pp. This paper concerns an invariant of metric spaces known as _magnitude_, which, as its name suggests, is a notion of size. It has its origins in category theory and was developed by Tom Leinster, though as it happens the same notion had been developed earlier by Andrew Solow and Stephen Polasky for completely different reasons: they were looking for a natural way to measure biodiversity. The connection between category theory and metric spaces is via the notion of an _enriched_ category, which is roughly speaking a category where the morphisms between two objects form not just a set but a set with additional structure. If $X$ is a metric space, then we can think of its elements as "objects" and for any two objects $x$ and $y$ we can regard a real number $r$ as a "morphism" if $r\geq d(x,y)$, using addition to compose morphisms. For composition to be possible, we need it to be the case that if $r\geq d(x,y)$ and $s\geq d(y,z)$, then $r+s\geq d(x,z)$, but that requirement is equivalent to the triangle inequality. (This is not how the definitions actually work, but it is a quick way of indicating how there could be a connection between categories and metric spaces.) It turns out that Euler characteristic can be generalized to finite categories and provides a fundamental notion of size. Pursuing this idea and generalizing it further to enriched categories led Leinster to formulate the following notion of magnitude for a finite metric space. Let $X$ be a finite metric space and define a matrix indexed by $X^2$ by taking its $xy$ entry to be $e^{-d(x,y)}$. Assuming that this matrix is invertible, then the magnitude of $X$ is the sum of the entries of the inverse. (In general, not all finite metric spaces have magnitude, though the definition can be extended to many finite metric spaces for which the above matrix is not invertible, including all finite subsets of Euclidean spaces.) There are two natural ways to generalize this notion from finite metric spaces to compact metric spaces. One is to consider discretizations, and the other is to replace matrices by kernels and sums by integrals. Mark Meckes showed that the two approaches give essentially the same results, so there is a well-established notion of magnitude for compact metric spaces as well. It appears that magnitude captures many other important metric notions. For instance, if $A$ is a compact subset of $\mathbb R^n$, one can look at the growth rate of the magnitude of $tA$. For many sets $A$, this function contains a great deal of interesting information about $A$ that appears to relate closely to existing well-known invariants, though this is an area that is still being actively investigated. This paper considers the case of balls in Euclidean space of odd dimension. An algorithm for calculating the magnitude was given by Juan Antonio Barceló and Tony Carbery, who proved that it was a rational function of the radius and calculated the function for a few low dimensions. For instance, they showed that the magnitude of a 5-dimensional ball of radius $r$ is $$\frac{r^6+18r^5+135r^4+535r^3+1080r^2+1080r+360}{5!(r+3)}.$$ It is difficult to look at such an expression without wondering whether the coefficients can be interpreted somehow -- in particular, it is notable that in the examples calculated by Barceló and Carbery they are all positive. This paper derives a formula for the magnitude in general dimension (as opposed to an algorithm for calculating the formula in any given dimension), which allows the author to show that the coefficients are indeed always positive, as well as giving them an interpretation. This is done using Hankel determinants of reverse Bessel polynomials, which are defined on page 3 of the paper, which then open the way to more combinatorial interpretations in terms of objects called weighted Schröder paths.

**The link is currently to the submitted version. The published version will appear very soon.** The structure of multiplicative functions with small partial sums, Discrete Analysis 2020:6, 19 pp. A _multiplicative function_ is a function from $\mathbb N$ to $\mathbb C$ with the property that $f(ab)=f(a)f(b)$ whenever $a$ and $b$ are coprime. Multiplicative functions arise very naturally and are of fundamental importance in number theory. From the definition (and the fundamental theorem of arithmetic) it follows that a multiplicative function is determined by the values it takes on prime powers. A particularly important example is the Möbius function $\mu$. This takes the value 1 at 1, -1 at all primes $p$, and 0 at all other prime powers. From this it follows that $\mu(n)$ is zero if $n$ has a repeated prime factor and is otherwise 1 if the number of prime factors of $n$ is even and -1 if the number of prime factors is odd. An indication of the importance of $\mu$ is that the Riemann hypothesis is equivalent to the assertion that $|\sum_{n\leq x}\mu(n)|=x^{\frac 12+o(1)}$, or in other words, that the partial sum up to $x$ does not get much larger than $\sqrt x$ in magnitude. More generally, understanding the behaviour of partial sums of multiplicative functions pays significant dividends in number theory. A notable feature of the Möbius function is that it takes the same value at each prime. While many of the most commonly studied multiplicative functions do not have this property, they frequently have the weaker property that their _average_ along primes tends to a limit. It is therefore natural to ask what relationship there is between this limiting average, if it exists, and the partial sums. This question was addressed by Landau in 1909, and his work was developed by Selberg and Delange. The resulting Landau-Selberg-Delange method (often called the LSD method) provides an asymptotic formula for the partial sums in terms of its average along primes. From the asymptotic formula, it can be deduced that the partial sums are small (that is, significantly smaller than $x$) only if the average is a non-positive integer. For a multiplicative function to have small partial sums, it is not necessary for its average values along primes to converge. Consider for example the function $f(n)=\mu(n)n^{i\gamma}$ for some real number $\gamma$, which takes the value $-p^{i\gamma}$ at a prime $p$. The function $n^{i\gamma}=e^{i\gamma\log n}$ progresses round the unit circle at ever-decreasing speed, and the Möbius function, which behaves in a "random" way, provides a great deal of cancellation, so the partial sums of $f$ are still small. But because of the slow growth rate of $\log n$, the average of $f(p)$ up to $x$ does not converge. The purpose of this paper is to look at the considerably harder _inverse_ problem that arises from observations like this: suppose we know that the partial sums of a multiplicative function (satisfying suitable conditions) are small. What can we say about its behaviour along the primes? The result is that, provided the multiplicative function satisfies a certain not very stringent condition, then we must have on average that $$f(p)\approx -p^{i\gamma_1}-p^{i\gamma_2}-\dots-p^{i\gamma_k}$$ for some real numbers $\gamma_1,\dots,\gamma_k$. More precisely, we must have that $$\lim_{x\to\infty} x^{-1}\sum_{p\leq x}(f(p)+p^{i\gamma_1}+p^{i\gamma_2}+\dots+p^{i\gamma_k})\log p=0.$$ Moreover, if $|f(p)|\leq D$ for every $p$, then $k$ can be chosen to be at most $D$. (Note that each $p^{i\gamma_j}$ has modulus 1, so this is on average necessary as well.) It is reasonably straightforward to prove that if $f$ satisfies a condition like this, then its partial sums are small, so this gives a very nice characterization of such functions. Note that if we set $k=1$ and $\gamma_1=0$, then $f(p)$ averages -1, so $f$ is in some sense close to the Möbius function. This paper generalizes work of the first author, who showed the result when $D=1$. That is, the conclusion holds with $k=0$ or $1$ if $|f(p)|\leq 1$ for every $p$.

We give a counterexample to a trilinear version of the operator space Grothendieck theorem. In particular, we show that for trilinear forms on $\ell_\infty$, the ratio of the symmetrized completely bounded norm and the jointly completely bounded norm is in general unbounded, answering a question of Pisier. The proof is based on a non-commutative version of the generalized von Neumann inequality from additive combinatorics.

On the Lehmer conjecture and counting in finite fields, Discrete Analysis 2019:5, 8pp. Let $\alpha$ be an algebraic number and let $a_0\prod_{i=1}^k(x-\alpha_i)$ be its minimal polynomial. The _Mahler measure_ of $\alpha$ is defined to be the quantity $|a_0|\prod_{i=1}^k\max\{1,|\alpha_i|\}$. For example, $\sqrt 2$ has minimal polynomial $x^2-2=(x-\sqrt 2)(x+\sqrt 2)$, so its Mahler measure is $2$, and if $\alpha$ is a root of unity, then its Mahler measure is 1. The Lehmer conjecture asserts that there exists an absolute constant $c_0>0$ such that every algebraic number that is _not_ a root of unity has Mahler measure at least $1+c_0$. The smallest known Mahler measure greater than 1 comes from the polynomial $$x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1,$$ which has exactly one root outside the unit disc (making that root a so-called _Salem number_), which is approximately equal to $1.176280818$. Thus, $0.176280819$ is an upper bound for $c_0$, if it exists. Let $S_d$ be the set of all polynomials of degree at most $d$ with coefficients in $\{0,1\}$. In an earlier paper [1], the authors of this paper proved that the Lehmer conjecture is equivalent to the assertion that the growth rate of the set $\{P(\alpha):P\in S_d\}$ as $d$ tends to infinity is at least $(1+c_1)^d$ for some absolute constant $c_1>0$, again as long as $\alpha$ is not a root of unity. (If it is a root of unity, then the growth rate is polynomial.) The purpose of this short paper is to present a further equivalence. The authors define a prime $p$ to be $C$-_wild_ if there exists some $x$ of multiplicative order at least $(\log p)^2$ such that the $(\log p)^C$-fold sumset of the geometric progression $H=\{x,x^2,\dots,x^{|C\log p|}\}$ is not the whole of $\mathbb F_p$. They then prove that the Lehmer conjecture is true if and only if there exists $C$ such that the proportion of primes less than $n$ that are $C$-wild tends to zero as $n$ tends to infinity. The "if" direction of this equivalence turns out not to be too hard: with the help of well-known results, an algebraic number $\alpha$ with Mahler measure close to 1 can be used to show that there is a constant $C$ and a dense set of $C$-wild primes. For this reason, the authors do not claim that counting wild primes is likely to be easier than proving the Lehmer conjecture directly. The fact that the Lehmer conjecture is related to additive combinatorics has been observed already, as the authors acknowledge, but the connection is particularly cleanly expressed here. The principal novelty of the paper is the converse statement. It is quite surprising that a well-known conjecture in algebraic number theory is actually equivalent to a simple counting problem in $\mathbb F_p$: this sheds new light on the conjecture and perhaps helps to explain why it is difficult. [1] Breuillard, E. and Varjú, P. P., _Entropy of Bernoulli convolutions and uniform exponential growth for linear groups,_ J. Anal. Math., to appear. Also available at [arXiv:1510.04043](https://arxiv.org/abs/1510.04043)