The Erd\H{o}s distinct distances problem in $\mathbb{R}^3$
We prove that $N$ points in $\mathbb{R}^3$ determine at least $N^{2/3-o(1)}$ distinct distances.
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arXiv subjects
Jonathan Tidor
Publications and source records attributed to Jonathan Tidor.
$K_{2,t+1}$-free graphs with many copies of $K_{t,t}$
For every fixed integer $t\geq 3$, we construct an $n$-vertex $K_{2,t+1}$-free graph containing $\Omega_t(n^2)$ copies of $K_{t,t}$. Combined with a simple counting argument, this shows that \[ \mathrm{ex}(n,K_{t,t},K_{2,t+1})=\Theta_t(n^2). \] This answers a question of Spiro.
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Separators for intersection graphs of spheres
We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced separator of size $O_d(m^{1/d}n^{1-2/d})$. This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.
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On monochromatic solutions to linear equations over the integers
We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of $\{1,\ldots,n\}$. We further study commonness of four-term equations and disprove a conjecture of Costello and Elvin by showing that, unlike over $\mathbb{F}_p$, the four-term equation $x_1 + 2x_2 - x_3 - 2x_4 = 0$ is uncommon over $\{1,\ldots,n\}$.
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More unit distances in arbitrary norms
For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon, Buci\'c, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for $d\geq 3$ and a typical norm on $\mathbb R^d$, the unit distance graph of this norm contains a copy of $K_{d,m}$ for all $m$.
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Multilevel polynomial partitioning and semialgebraic hypergraphs: regularity, Tur\'an, and Zarankiewicz results
We prove three main results about semialgebraic hypergraphs. First, we prove an optimal and oblivious regularity lemma. Fox, Pach, and Suk proved that the class of $k$-uniform semialgebraic hypergraphs satisfies a very strong regularity lemma where the vertex set can be partitioned into $\mathrm{poly}(1/\varepsilon)$ parts so that all but an $\varepsilon$-fraction of $k$-tuples of parts are homogeneous (either complete or empty). Our result improves the number of parts in the partition to $O_{d,k}((D/\varepsilon)^{d})$ where $d$ is the dimension of the ambient space and $D$ is a measure of the complexity of the hypergraph; additionally, the partition is oblivious to the edge set of the hypergraph. We give examples that show that the dependence on both $\varepsilon$ and $D$ is optimal. From this regularity lemma we deduce the best-known Tur\'an-type result for semialgebraic hypergraphs. Third, we prove a Zarankiewicz-type result for semialgebraic hypergraphs. Previously Fox, Pach, Sheffer, Suk, and Zahl showed that a $K_{u,u}$-free semialgebraic graph on $N$ vertices has at most $O_{d,D,u}(N^{2d/(d+1)+o(1)})$ edges and Do extended this result to $K_{u,\ldots,u}^{(k)}$-free semialgebraic hypergraphs. We improve upon both of these results by removing the $o(1)$ in the exponent and making the dependence on $D$ and $u$ explicit and polynomial. All three of these results follow from a novel ``multilevel polynomial partitioning scheme'' that efficiently partitions a point set $P\subset\mathbb{R}^d$ via low-complexity semialgebraic pieces. We prove this result using the polynomial method over varieties as developed by Walsh which extends the real polynomial partitioning technique of Guth and Katz. We give additional applications to the unit distance problem, the Erd\H{o}s--Hajnal problem for semialgebraic graphs, and property testing of semialgebraic hypergraphs.
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Ramsey and Tur\'an numbers of sparse hypergraphs
Degeneracy plays an important role in understanding Tur\'an- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a $k$-uniform hypergraph as the degeneracy of its $1$-skeleton (i.e., the graph formed by replacing every $k$-edge by a $k$-clique). We prove that skeletal degeneracy controls hypergraph Tur\'an and Ramsey numbers in a similar manner to (graphical) degeneracy. Specifically, we show that $k$-uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erd\H{o}s conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Tur\'an number of a $k$-uniform $k$-partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process' introduced by Lee in his resolution of the Burr-Erd\H{o}s conjecture.
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Triangle Ramsey numbers of complete graphs
A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}{3}\] for all sufficiently large $t$. We do so through a result on graph coloring: there exists an absolute constant $K$ such that every $r$-chromatic graph where every edge is contained in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles in total.
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Uniform sets with few progressions via colorings
Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-AP) density at most $\alpha^C$, for arbitrarily large $C$. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c>0$. We show that an affirmative answer to Ruzsa's question would follow from the existence of an $N^{o(1)}$-coloring of $[N]$ without symmetrically colored 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa's question is equivalent to our arithmetic Ramsey question. We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and $k$-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalizations to arbitrary one-dimensional patterns.
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Cubic Goldreich-Levin
In this paper, we give a cubic Goldreich-Levin algorithm which makes polynomially-many queries to a function $f \colon \mathbb F_p^n \to \mathbb C$ and produces a decomposition of $f$ as a sum of cubic phases and a small error term. This is a natural higher-order generalization of the classical Goldreich-Levin algorithm. The classical (linear) Goldreich-Levin algorithm has wide-ranging applications in learning theory, coding theory and the construction of pseudorandom generators in cryptography, as well as being closely related to Fourier analysis. Higher-order Goldreich-Levin algorithms on the other hand involve central problems in higher-order Fourier analysis, namely the inverse theory of the Gowers $U^k$ norms, which are well-studied in additive combinatorics. The only known result in this direction prior to this work is the quadratic Goldreich-Levin theorem, proved by Tulsiani and Wolf in 2011. The main step of their result involves an algorithmic version of the $U^3$ inverse theorem. More complications appear in the inverse theory of the $U^4$ and higher norms. Our cubic Goldreich-Levin algorithm is based on algorithmizing recent work by Gowers and Mili\'cevi\'c who proved new quantitative bounds for the $U^4$ inverse theorem. Our cubic Goldreich-Levin algorithm is constructed from two main tools: an algorithmic $U^4$ inverse theorem and an arithmetic decomposition result in the style of the Frieze-Kannan graph regularity lemma. As one application of our main theorem we solve the problem of self-correction for cubic Reed-Muller codes beyond the list decoding radius. Additionally we give a purely combinatorial result: an improvement of the quantitative bounds on the $U^4$ inverse theorem.
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Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields
This paper gives the first quantitative bounds for the inverse theorem for the Gowers $U^4$-norm over $\mathbb{F}_p^n$ when $p=2,3$. We build upon earlier work of Gowers and Mili\'cevi\'c who solved the corresponding problem for $p\geq 5$. Our proof has two main steps: symmetrization and integration of low-characteristic trilinear forms. We are able to solve the integration problem for all $k$-linear forms, but the symmetrization problem we are only able to solve for trilinear forms. We pose several open problems about symmetrization of low-characteristic $k$-linear forms whose resolution, combined with recent work of Gowers and Mili\'cevi\'c, would give quantitative bounds for the inverse theorem for the Gowers $U^{k+1}$-norm over $\mathbb{F}_p^n$ for all $k,p$.
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Non-classical polynomials and the inverse theorem
In this note we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm. We give a brief deduction of the fact that a bounded function on $\mathbb F_p^n$ with large $U^k$-norm must correlate with a classical polynomial when $k\leq p+1$. To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm over $\mathbb F_p^n$ for all $k\geq p+2$, completely characterizing when classical polynomials suffice.
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Popular differences for matrix patterns
The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ be a finite abelian group of order $N$, and $A\subseteq G^k$ with $|A|\ge\alpha N^k$. If $M_1$, $M_2$, $M_1-M_2$, and $M_1+M_2$ are automorphisms of $G^k$, is it true that there exists a popular difference $d \in G^k\setminus\{0\}$ such that \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (\alpha^4-o(1))N^k.\] We show that this conjecture is false in general, but holds for $G = \mathbb{F}_p^n$ with $p$ an odd prime given the additional spectral condition that no pair of eigenvalues of $M_1M_2^{-1}$ (over $\overline{\mathbb{F}}_p$) are negatives of each other. In particular, the "rotated squares" pattern does not satisfy this eigenvalue condition, and we give a construction of a set of positive density in $(\mathbb{F}_5^n)^2$ for which that pattern has no nonzero popular difference. This is in surprising contrast to three-point patterns, which we handle over all compact abelian groups and which do not require an additional spectral condition.
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Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost
We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of $w$ workers and a multiset $T \subseteq[t]$ of $|T|=w$ tasks, a memoryless worker-task assignment function is any function $\phi$ that assigns the workers $[w]$ to the tasks $T$ based only on the current value of $T$. The assignment function $\phi$ is said to have switching cost at most $k$ if, for every task multiset $T$, changing the contents of $T$ by one task changes $\phi(T)$ by at most $k$ worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost. In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sub-linear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020). We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost $O(\log w \log (wt))$ and an explicit construction that achieves switching cost $\operatorname{polylog} (wt)$. We also prove a super-constant lower bound on switching cost: we show that for any value of $w$, there exists a value of $t$ for which the optimal switching cost is $w$. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of $w$. Finally, we present an application of the worker-task assignment problem to a metric embeddings problem. In particular, we use our results to give the first low-distortion embedding from sparse binary vectors into low-dimensional Hamming space.
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Joints of varieties
We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of $N$ planes is replaced by a set of 2-dimensional algebraic varieties of total degree $N$, and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety.
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Hypergraph expanders of all uniformities from Cayley graphs
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.
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Spherical two-distance sets and eigenvalues of signed graphs
We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{\alpha,\beta}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{\alpha,\beta\}$. For fixed $-1\leq\beta<0\leq\alpha<1$, we propose a conjecture for the limit of $N_{\alpha,\beta}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $\alpha+2\beta<0$ or $(1-\alpha)/(\alpha-\beta) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $\alpha = -\beta$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{\alpha,\beta}(d)/d$ for any nontrivial fixed values of $\alpha$ and $\beta$ outside of the equiangular lines setting.
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A counterexample to the Bollob\'as-Riordan conjectures on sparse graph limits
Bollob\'as and Riordan, in their paper "Metrics for sparse graphs," proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom $C_4$-counts), but with no limit expressible as a kernel.
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