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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., Comment: 32 pages

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., Comment: 40 Pages

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 Laba and Chen required randomness and only covered the range $0 < b \leq a \leq d=1$. We also resolve a question of Seeger about the Fourier restriction inequality on the sets of well-approximable numbers., Comment: 27 Pages

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., Comment: 37 Pages

We obtain a Fourier dimension estimate for sets of exact approximation order introduced by Bugeaud for certain approximation functions $\psi$. This Fourier dimension estimate implies that these sets of exact approximation order contain normal numbers., Comment: 16 pages, one figure

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., Comment: 14 Pages

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., Comment: 32 Pages

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., Comment: 18 pages

Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $\alpha\in (0, 1)$. 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., Comment: 16 pages

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., Comment: 23 pages