Binary Quadratic Forms in Difference Sets

We show that if $h(x,y)=ax^2+bxy+cy^2\in \mathbb{Z}[x,y]$ satisfies $\Delta(h)=b^2-4ac\neq 0$, then any subset of $\{1,2,\dots,N\}$ lacking nonzero differences in the image of $h$ has size at most a constant depending on $h$ times $N\exp(-c\sqrt{\log N})$, where $c=c(h)>0$. We achieve this goal by adapting an $L^2$ density increment strategy previously used to establish analogous results for sums of one or more single-variable polynomials. Our exposition is thorough and self-contained, in order to serve as an accessible gateway for readers who are unfamiliar with previous implementations of these techniques.Comment: 14 pages, typos corrected, to appear in Proceedings of Combinatorial and Additive Number Theory 201

Group actions and a multi-parameter Falconer distance problem

In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${\textbf{d}}=(d_1,d_2, \dots, d_{\ell})\in \mathbb{N}^{\ell}$ with $d_1+d_2+\dots+d_{\ell}=d$ and $E \subseteq \mathbb{R}^d$, we define $$\Delta_{{\textbf{d}}}(E) = \left\{ \left(|x^{(1)}-y^{(1)}|,\ldots,|x^{(\ell)}-y^{(\ell)}|\right) : x,y \in E \right\} \subseteq \mathbb{R}^{\ell},$$ where for $x\in \mathbb{R}^d$ we write $x=\left( x^{(1)},\dots, x^{(\ell)} \right)$ with $x^{(i)} \in \mathbb{R}^{d_i}$. We ask how large does the Hausdorff dimension of $E$ need to be to ensure that the $\ell$-dimensional Lebesgue measure of $\Delta_{{\textbf{d}}}(E)$ is positive? We prove that if $2 \leq d_i$ for $1 \leq i \leq \ell$, then the conclusion holds provided $$\dim(E)>d-\frac{\min d_i}{2}+\frac{1}{3}.$$ We also note that, by previous constructions, the conclusion does not in general hold if $$\dim(E)<d-\frac{\min d_i}{2}.$$ A group action derivation of a suitable Mattila integral plays an important role in the argument

A Quantitative Result on Diophantine Approximation for Intersective Polynomials

In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomials structures in sumsets and limitations of the method.Comment: 6 page

Polynomial Differences in the Primes

We establish, utilizing the Hardy-Littlewood Circle Method, an asymptotic formula for the number of pairs of primes whose differences lie in the image of a fixed polynomial. We also include a generalization of this result where differences are replaced with any integer linear combination of two primes.Comment: References added, some typos correcte

Difference Sets and Polynomials

We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in \mathbb{Z}[x]$ lie in in the classes of so-called intersective and $\mathcal{P}$-intersective polynomials, respectively. For example, we show that a subset of $\{1,2,\dots,N\}$ free of nonzero differences of the form $n^j+m^k$ for fixed $j,k\in \mathbb{N}$ has density at most $e^{-(\log N)^{\mu}}$ for some $\mu=\mu(j,k)>0$. Our results, obtained by adapting two Fourier analytic, circle method-driven strategies, either recover or improve upon all previous results for a single polynomial. UPDATE: While the results and proofs in this preprint are correct, the main result (Theorem 1.1) has been superseded prior to publication by a new paper ( https://arxiv.org/abs/1612.01760 ) that provides better results with considerably less technicality, to which the interested reader should refer.Comment: 31 pages. The interested reader should likely instead refer to arXiv:1612.0176

Polynomials and Primes in Generalized Arithmetic Progressions (Revised Version)

We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate polynomials h with integer coefficients. The prime variant can be interpreted as a multi-dimensional, polynomial extension of Linnik's Theorem. This version is a revision of the published version. Most notably, the properness hypotheses have been removed from Theorems 2 and 3, and the numerology in Theorem 2 has been improved.Comment: 14 pages, typos corrected, numerology improved, properness hypotheses eliminate

A Purely Combinatorial Approach to Simultaneous Polynomial Recurrence Modulo 1

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.Comment: 6 page

Sets in Rd\mathbb{R}^d with slow-decaying density that avoid an unbounded collection of distances

For any $d\in \mathbb{N}$ and any function $f:(0,\infty)\to [0,1]$ with $f(R)\to 0$ as $R\to \infty$, we construct a set $A \subseteq \mathbb{R}^d$ and a sequence $R_n \to \infty$ such that $\|x-y\| \neq R_n$ for all $x,y\in A$ and $\mu(A\cap B_{R_n})\geq f(R_n)\mu(B_{R_n})$ for all $n\in \mathbb{N}$, where $B_R$ is the ball of radius $R$ centered at the origin and $\mu$ is Lebesgue measure. This construction exhibits a form of sharpness for a result established independently by Furstenberg-Katznelson-Weiss, Bourgain, and Falconer-Marstrand, and it generalizes to any metric induced by a norm on $\mathbb{R}^d$.Comment: 3 pages, minor typos correcte

Coinductive Invertibility in Higher Categories

Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have desirable properties. We define some properties we expect to hold in any reasonable definition of a weak $\omega$-category. With these properties we define three notions of invertibility inspired by homotopy type theory. These are quasi-invertibility, where a two sided inverse is required, bi-invertibility, where a separate left and right inverse is given, and half-adjoint inverse, which is a quasi-inverse with an extra coherence condition. These definitions take the form of coinductive data structures. Using coinductive proofs we are able to show that these three notions are all equivalent in that given any one of these invertibility structures, the others can be obtained. The methods used to do this are generic and it is expected that the results should be applicable to any reasonable model of higher category theory. Many of the results of the paper have been formalised in Agda using coinductive records and the machinery of sized types

Computation on Elliptic Curves with Complex Multiplication

We give the complete list of possible torsion subgroups of elliptic curves with complex multiplication over number fields of degree 1-13. Additionally we describe the algorithm used to compute these torsion subgroups and its implementation.Comment: 24 pages, 3 figures, submitte