Related papers: A discretised projection theorem in the plane

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider the Riemann zeta function $\zeta(s)$ in the $t$-aspect, Dirichlet $L$-functions in…

Utilising recent advances in incidence geometry for balls and tubes, and advances in sum-product theory in the discrete setting, we show that for $0 < s \leq 1/2$ and for any $A \subset \mathbb{R}$ with Hausdorff dimension $s$, either the…

In a projective plane $\Pi_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\mathcal{S}$ is saturating (or dense) if any point of $\Pi_{q}\setminus \mathcal{S}$ is collinear with two points in $\mathcal{S}$. Modifying an…

We prove a lower bound on the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on planar open sets, in terms of their inradius and topology. The result is optimal, in many respects. In particular, we recover a classical…

Let $S(t) = \frac{1}{\pi}\Im \log\zeta\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log…

Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. An $(s,t)$-Furstenberg set is a set $K \subset \mathbb{R}^{2}$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension $\dim_{\mathrm{H}} \mathcal{L} \geq t$…

Let $p$ be an odd prime and let $E\subset \mathbb{F}_p^2$ with $|E|=p^a$, where $0<a\le 1$. For a direction $V$ (a $1$-dimensional subspace of $\mathbb{F}_p^2$), let $\pi^V:\mathbb{F}_p^2\to \mathbb{F}_p^2/V$ denote the quotient map. We…

We prove the following restricted projection theorem. Let $n\ge 3$ and $\Sigma \subset S^{n}$ be an $(n-1)$-dimensional $C^2$ manifold such that $\Sigma$ has sectional curvature $>1$. Let $Z \subset \mathbb{R}^{n+1}$ be analytic and let $0…

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every non-trivial measurable set $S$ of real numbers, there is a point at which both the lower densities of $S$ and of the complement of $S$ are at least…

(I) We exhibit a set of 23 points in the plane that has dilation at least $1.4308$, improving the previously best lower bound of $1.4161$ for the worst-case dilation of plane spanners. (II) For every integer $n\geq13$, there exists an…

We fully resolve the Furstenberg set conjecture in $\mathbb{R}^2$, that a $(s, t)$-Furstenberg set has Hausdorff dimension $\ge \min(s+t, \frac{3s+t}{2}, s+1)$. As a result, we obtain an analogue of Elekes' bound for the discretized…

We show that a set $A \subset \{0,1\}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + \epsilon)$ can be made into a subcube by at most $(2 \epsilon/\log_{2}(1/\epsilon))|A|$ additions and deletions, provided $\epsilon$…

In this paper we prove that every set $A\subset\mathbb{Z}$ satisfying the inequality $\sum_{x}\min(1_A*1_A(x),t)\le(2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use…

Given a compact subset $\Sigma \subset \mathbb{R}$ (or $\mathbb{C}$) with logarithmic capacity greater than zero, we construct an explicit family of probability measures supported on $\Sigma$ such that their closure is all the possible weak…

Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(\gamma(\theta),\gamma'(\theta),\gamma''(\theta)\big)\neq 0$. For each $\theta\in[0,1]$, let…

We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ \Omega=\{\xi\in \mathbb{R}: \dim_B(E+\xi E)=\dim_B E\}. $$ We prove that either $\Omega=\{0\}$ or $\Omega$ is a…

Let A be a finite subset of a commutative additive group Z. The sumset and difference set of A are defined as the sets of pairwise sums and differences of elements of A, respectively. The well-known inequality $\sigma(A)^{1/2} \leq…

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving…

Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0 and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1, 0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ having…

Given $0 < s < 1$, I prove that there exists a constant $\epsilon = \epsilon(s) > 0$ such that the following holds. Let $K \subset \mathbb{R}^{2}$ be a Borel set with $\mathcal{H}^{1}(K) > 0$, and let $E_{s}(K) \subset S^{1}$ be the…