arXiv subjects
Publications and source records attributed to Audie Warren.
Adapting the construction disproving the sum-product conjecture over $\mathbb R$ present in Bloom, Sawin, Schildkraut and Zhelezov, we show the existence of a constant $c>0$ and arbitrarily large finite sets $A \subseteq \mathbb R$ such that $$|AA+A+A| \ll |A|^{2-c}.$$ As a corollary, all of the sets $A+A$, $AA$, $(A+1)(A+1)$, $A(A+1)$ and $AA+A$ are of size $O(|A|^{2-c})$ for this construction.
A one-degree-of-freedom graph is a graph obtained from a minimally rigid graph in the plane and removing an edge. For such graph, the set of realisations with fixed edge length, modulo rotations and reflections, is an algebraic curve. The genus of a connected component for generic edge lengths is a number that depends only on the graph. We prove that this genus is always odd, unless it is zero. The proof is based on tropical geometry.
A graph is said to be rigid if, given a generic realisation of the graph as a bar-and-joint framework in the plane, there exist only finitely many other realisations of the graph with the same edge lengths modulo rotations, reflections and translations. In recent years there has been an increase of interest in determining exactly what this finite amount is, hereon known as the realisation number. Combinatorial algorithms for the realisation number were previously known for the special cases of minimally rigid and redundantly rigid graphs. In this paper we provide a combinatorial algorithm to compute the realisation number of any rigid graph, and thus solve an open problem of Jackson and Owen. We then adapt our algorithm to compute: (i) spherical realisation numbers, and (ii) the number of rank-3 PSD matrix completions of a generic partial matrix.
In this paper we study the number of four-rich points defined by pencils of certain algebraic objects. Our main result concerns the number of four-rich points defined by four sheaves of planes; under certain non-degeneracy conditions, we prove that four sheaves of $n$ planes in $\mathbb P^3$ determine at most $O(n^{8/3})$ four-rich points. We prove this using the four dimensional Elekes-Szab\'{o} theorem. Using the same method, we prove an upper bound on the number of four-rich points determined by four sets of concentric spheres in $\mathbb C^3$. Furthermore, using the same technique with the 3-d Elekes-Szab\'{o} theorem, one can prove upper bounds on four-rich points determined by various configurations of lines/circles in the plane $\mathbb C^2$; we give one such example, involving two pencils of lines and two pencils of concentric circles in $\mathbb C^2$.
The famous Erd\H{o}s distinct distances problem asks the following: how many distinct distances must exist between a set of $n$ points in the plane? There are many generalisations of this question that ask one to consider different spaces and metrics, or larger structures of points. We bring these problems into a common framework using the concept of $g$-rigidity. Specifically, if $G=(V,E)$ is a (hyper)graph, $g$ is a map assigning polynomial measurements to the edges of $G$ and $f_{g,G}(P^V)$ gives the set of $g$-distinct realisations of the $g$-rigid graph $G$, where vertices must lie in a point set $P$, our main results describe sharp lower bounds for the size of $\big|f_{g,G}(P^V)\big|$. This allows us to obtain results for pseudo-Euclidean metrics, $\ell_p$ metrics, dot-product problems, matrix completion problems, and symmetric tensor completion problems. In addition, we use the recent work of Alon, Buci\'c and Sauermann along with a simple colouring argument to prove that the number of $\| \cdot\|$-distinct realisations of a graph $G=(V,E)$ within a $d$-dimensional point set $P$ is at least $\Omega\left(\frac{|P|^{|V|-1}}{(\log |P|)^2} \right)$ for almost all $d$-norms. Our methods here also provide a short proof that the unit distance conjecture implies the pinned distance conjecture.
In this paper we investigate which subsets of the real plane are realisable as the set of points on which a one-layer ReLU neural network takes a positive value. In the case of cones we give a full characterisation of such sets. Furthermore, we give a necessary condition for any subset of $\mathbb R^d$. We give various examples of such one-layer neural networks.
This paper focuses on studying the configuration spaces of graphs realised in $\mathbb C^2$, such that the configuration space is, after normalisation, one dimensional. If this is the case, then the configuration space is, generically, a smooth complex curve, and can be seen as a Riemann surface. The property of interest in this paper is the genus of this curve. Using tropical geometry, we give an algorithm to compute this genus. We provide an implementation in Python and give various examples.
For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the corresponding edge label. This set of functions can be expressed as the zero set of quadratic polynomials and our main result characterizes the number of complex irreducible components of this zero set in terms of combinatorial properties of the graph. In case the complex components are three-dimensional, then the graph is minimally rigid and the component number is a well-known invariant from rigidity theory. If the components are four-dimensional, then they correspond to one-dimensional coupler curves of flexible planar mechanisms. As an application, we characterize the degree of irreducible components of such coupler curves combinatorially.
For every graph that is mimimally rigid in the plane, its Galois group is defined as the Galois group generated by the coordinates of its planar realizations, assuming that the edge lengths are transcendental and algebraically independent. Here we compute the Galois group of all minimally rigid graphs that can be constructed from a single edge by repeated Henneberg 1-steps. It turns out that any such group is totally imprimitive, i.e., it is determined by all the partitions it preserves.
In this paper we prove an incidence bound for points and cubic curves over prime fields. The methods generalise those used by Mohammadi, Pham, and Warren (2021).
We construct a convex set $A$ with cardinality $2n$ and with the property that an element of the difference set $A-A$ can be represented in $n$ different ways. We also show that this construction is optimal by proving that for any convex set $A$, the maximum possible number of representations an element of $A-A$ can have is $\lfloor |A|/2 \rfloor $.
In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck's theorem for conics.
We develop the methods used by Rudnev and Wheeler to prove an incidence theorem between arbitrary sets of Möbius transformations and point sets in $\mathbb F_p^2$. We also note some asymmetric incidence results, and give applications of these results to various problems in additive combinatorics and discrete geometry.
We give a construction of a set $A \subset \mathbb N$ such that any subset $A' \subset A$ with $|A'| \gg |A|^{2/3}$ is neither an additive nor multiplicative Sidon set. In doing so, we refute a conjecture of Klurman and Pohoata.
In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$ This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to also prove the three-variable expansion result \[ |AB+A|\gtrsim |A|^{\frac32 +\frac3{170}}\,. \] Our methods follow a series of recent developments in the sum-product literature, presenting a unified picture. Of particular interest is an adaptation of a regularisation technique of Xue, that enables us to find positive proportion subsets with certain desirable properties.
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 $F\in \mathbb{Q}[x,y,z]$ belonging to a particular family of polynomials, and any finite sets $A, B, C \subset \mathbb Q$ with $|A|=|B|=|C|=n$, we have \[ |Z(F) \cap (A\times B \times C)| \ll n^{2-\frac{1}{s}}. \] The value of the integer $s$ is dependent on the polynomial $F$, but is always bounded by $s \leq 5$, and so even in the worst applicable case this gives a quantitative improvement on a bound of Raz, Sharir and de Zeeuw (arXiv:1504.05012). We give several applications to problems in discrete geometry and arithmetic combinatorics. For instance, for any set $P \subset \mathbb Q^2$ and any two points $p_1,p_2 \in \mathbb Q^2$, we prove that at least one of the $p_i$ satisfies the bound \[ | \{ \| p_i - p \| : p \in P \}| \gg |P|^{3/5}, \] where $\| \cdot \|$ denotes Euclidean distance. This gives a conditional improvement to a result of Sharir and Solymosi (arXiv:1308.0814).
An arc is a subset of $\mathbb F_q^2$ which does not contain any collinear triples. Let $A(q,k)$ denote the number of arcs in $\mathbb F_q^2$ with cardinality $k$. This paper is primarily concerned with estimating the size of $A(q,k)$ when $k$ is relatively large, namely $k=q^t$ for some $t>0$. Trivial estimates tell us that \[ {q \choose k} \leq A(q,k) \leq {q^2 \choose k}. \] We show that the behaviour of $A(q,k)$ changes significantly close to $t=1/2$. Below this threshold an elementary argument is used to prove that the trivial upper bound above cannot be improved significantly. On the other hand, for $t \geq 1/2+δ$, we use the theory of hypergraph containers to get an improved upper bound \[ A(q,k) \leq {q^{2-t+2δ} \choose k}. \] This technique is also used to give an upper bound for the size of the largest arc in a random subset of $\mathbb F_q^2$ which holds with high probability. For example, we prove that a $p$-random subset $Q \subset \mathbb F_q^2$ with $q^{-3/2}<p<q^{-1}$ contains an arc of size $Ω(q^{1/2})$ with high probability. The result is optimal for this range of $p$. Finally, this optimal bound for arcs in random sets is used to prove a finite field analogue of a result of Balogh and Solymosi, with a better exponent: there exists a subset $P \subset \mathbb F_q^2$ which does not contain any collinear quadruples, but with the property that for every $P' \subset P$ with $|P'| \geq |P|^{3/4+o(1)}$, $P'$ contains a collinear triple.
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 affine group, a quantitative version of a theorem by Elekes about rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P, there may be at most one line, meeting the set of lines defined by P in at most a constant multiple of |P| points.