arXiv subjects
Publications and source records attributed to Robert Fraser.
We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions $d$ and the full parameter range $0 < a,b < d$. Our construction is deterministic and also yields Salem sets.
We consider measures supported on sets of irrational numbers possessing many consecutive partial quotients satisfying a condition based on the previous partial quotients. We show that under mild assumptions, such sets will always support measures whose Fourier transform decays to zero.
We construct a measure on the well-approximable numbers whose Fourier transform decays at a nearly optimal rate. This gives a logarithmic improvement on a previous construction of Kaufman.
We develop a theory of oscillatory integrals whose phase is given by the trace of a polynomial over an algebraic number field. We present an application to the singular integral for a version of Tarry's problem for algebraic integers.
We use a deterministic construction to prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $d=1$ and parameter range $0 < a,b \leq d$ and $b\leq 2a$. Previous constructions by Hambrook and {\L}aba \cite{HL2013} and Chen \cite{chen} required randomness and only covered the range $0 < b \leq a \leq d=1$. We also resolve a question of Seeger \cite{seeger-private} about the Fourier restriction inequality on the sets of well-approximable numbers.
We show for decreasing, positive approximation functions $\psi$ such that $\tau = \lim_{q \to \infty} \frac{\log \psi(q)}{\log q} < \frac{13 + \sqrt{73}}{8}$ and such that $q^2 \psi(q) \to 0$ that the set $\text{Exact}(\psi)$ of numbers approximable to the exact order $\psi$ has positive Fourier dimension. This implies that the set $\text{Exact}(\psi)$ contains normal numbers.
We obtain a Fourier dimension estimate for sets of exact approximation order introduced by Bugeaud for certain approximation functions $ψ$. This Fourier dimension estimate implies that these sets of exact approximation order contain normal numbers.
We construct the first explicit (i.e., non-random) examples of Salem sets in $\mathbb{R}^n$ of arbitrary prescribed Hausdorff dimension. This completely resolves a problem proposed by Kahane more than 60 years ago. The construction is based on a form of Diophantine approximation in number fields.
We show that subsets of $\mathbb{F}_q^{\infty}$ of large Fourier dimension must contain three-term arithmetic progressions. This contrasts with a construction of Shmerkin of a subset of $\mathbb{R}$ of Fourier dimension $1$ with no three-term arithmetic progressions.
In this paper we give an elementary proof of the local sum conjecture in two dimensions. In a remarkable paper [CMN, arXiv:1810.11340], this conjecture has been established in all dimensions using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture which will be accessbile to a broad readership.
Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $α\in (0, 1)$. Let $P: \mathbb{R}\to \mathbb{R}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Laba and the third author to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot, Laba and the third author.
For certain families of functions $\{f_q\}$ mapping $K^{nv_q} \to K^m$, where $K$ is a complete, nonarchimedean local field, we find a set $E$ of large Hausdorff dimension with the property that $f_q(x_1, \ldots, x_{v_q})$ is nonzero for any distinct points $x_1, \ldots, x_{v_q} \in E$. In particular, this result can be applied to show that the ring of integers of any local field contains a subset of Hausdorff dimension $1$ not containing any nondegenerate 3-term arithmetic progressions.
We discuss a framework for constructing large subsets of $\mathbb{R}^n$ and $K^n$ for non-archimedean local fields $K$. This framework is applied to obtain new estimates for the Hausdorff dimension of angle-avoiding sets and to provide a counterexample to a limiting version of the Capset problem.
We exhibit the first explicit examples of Salem sets in $\mathbb{Q}_p$ of every dimension $0 < α< 1$ by showing that certain sets of well-approximable $p$-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the $p$-adic setting.
We construct subsets of Euclidean space of large Hausdorff dimension and full Minkowski dimension that do not contain nontrivial patterns described by the zero sets of functions. The results are of two types. Given a countable collection of $v$-variate vector-valued functions $f_q : (\mathbb{R}^{n})^v \to \mathbb{R}^m$ satisfying a mild regularity condition, we obtain a subset of $\mathbb{R}^n$ of Hausdorff dimension $\frac{m}{v-1}$ that avoids the zeros of $f_q$ for every $q$. We also find a set that simultaneously avoids the zero sets of a family of uncountably many functions sharing the same linearization. In contrast with previous work, our construction allows for non-polynomial functions as well as uncountably many patterns. In addition, it highlights the dimensional dependence of the avoiding set on $v$, the number of input variables.
We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space $K^d$ over a local field with finite residue field. The construction is an adaptation of the ideas appearing in [12] and [13]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one presented in [4] over $\mathbb{F}_q[[t]]$.
Consider a sliding camera that travels back and forth along an orthogonal line segment $s$ inside an orthogonal polygon $P$ with $n$ vertices. The camera can see a point $p$ inside $P$ if and only if there exists a line segment containing $p$ that crosses $s$ at a right angle and is completely contained in $P$. In the minimum sliding cameras (MSC) problem, the objective is to guard $P$ with the minimum number of sliding cameras. In this paper, we give an $O(n^{5/2})$-time $(7/2)$-approximation algorithm to the MSC problem on any simple orthogonal polygon with $n$ vertices, answering a question posed by Katz and Morgenstern (2011). To the best of our knowledge, this is the first constant-factor approximation algorithm for this problem.