Acute sets of exponentially optimal size

Acute sets

A set of points in $\mathbb{R}^d$ is acute, if any three points from this set form an acute angle. In this note we construct an acute set in $\mathbb{R}^d$ of size at least $2^{d/2}$.

Acute sets

A set of points in $\mathbb{R}^d$ is acute, if any three points from this set form an acute triangle. In this note we construct an acute set in $\mathbb{R}^d$ of size at least $1.618^d$. Also, we present a simple example of an acute set of size at least $2^{\frac{d}{2}}$.

On metric convexity, the discrete Hahn-Banach theorem, separating systems and sets of points forming only acute angles

This text has three parts. The first one is largely autobiographical, hence my use of the first person. There I recall how Gerard Cohen influenced important parts of my research. The second is of a more classic mathematical nature. I present a discrete analogue of the Hahn-Banach theorem, which serves as a basis for generalizing the notion of separating systems in the context of metric convexity. The third one aims at building a bridge between two communities of researchers, those interested in separating systems, and those interested in a certain question in combinatorial geometry --- sets of…

关于实线上解析自共形集的指数分离

On exponential separation of analytic self-conformal sets on the real line

本文研究了实线上指数分离解析IFS的维度特性，证明其在C²拓扑下稠密，并引入双IFS工具，给出了指数分离的充分条件及构造方法。

绕过角落

Cutting corners

本文提出了一种结合指数Ramsey集的新方法，改进了禁止等边三角形的色数估计，并推导了其他组合问题的结论。

On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$

We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $Φ$ does not preserve a proper affine subspace, we show that one of the following holds: (1) the dimension is equal to the trivial bound (the minimum of $d$ and the similarity dimension $s$); (2) for all large $n$ there are $n$-fold compositions of maps from $Φ$ which are super-exponentially close in $n$; (3) there is a non-trivial linear subspace of $\mathbb{R}^{d}$ that is preserved by the linearization of $Φ$ and whose translates typically meet the set or measure in full dimensio…