Small subset sums

子集 Minkowski 和的体积与 Lyusternik 区域

Volumes of subset Minkowski sums and the Lyusternik region

论文引入并研究紧集族 Minkowski 子集和体积的 Lyusternik 区域，证明分数 BML 不等式在一维成立，并给出任意维度有效的变体。

Large convexly independent subsets of Minkowski sums

Let $E_d(n)$ be the maximum number of pairs that can be selected from a set of $n$ points in $R^d$ such that the midpoints of these pairs are convexly independent. We show that $E_2(n)\geq Ω(n\sqrt{\log n})$, which answers a question of Eisenbrand, Pach, Rothvoß, and Sopher (2008) on large convexly independent subsets in Minkowski sums of finite planar sets, as well as a question of Halman, Onn, and Rothblum (2007). We also show that $\lfloor\frac{1}{3}n^2\rfloor\leq E_3(n)\leq \frac{3}{8}n^2+O(n^{3/2})$. Let $W_d(n)$ be the maximum number of pairwise nonparallel unit distance pairs in a set o…

Sets of unit vectors with small subset sums

We say that a family ${x_i|i\in[m]}$ of vectors in a Banach space $X$ satisfies the $k$-collapsing condition if $|\sum_{i\in I}x_i|\leq 1$ for all $k$-element subsets $I\subseteq{1,2,...,m}$. Let $C(k,d)$ denote the maximum cardinality of a $k$-collapsing family of unit vectors in a $d$\dimensional Banach space, where the maximum is taken over all spaces of dimension $d$. Similarly, let $CB(k,d)$ denote the maximum cardinality if we require in addition that $\sum_{i=1}^m x_i=o$. The case $k=2$ was considered by Füredi, Lagarias and Morgan (1991). These conditions originate in a theorem of Lawl…

On arithmetic sums of Ahlfors-regular sets

Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions $\dim_{\mathrm{H}} A =: α$ and $\dim_{\mathrm{H}} B =: β$. I prove that $$\dim_{\mathrm{H}} [A + θB] \geq α+ β\cdot \tfrac{1 - α}{2 - α}$$ for all $θ\in \mathbb{R} \, \setminus \, E$, where $\dim_{\mathrm{H}} E = 0$.