Related papers: Distinct volume subsets

For positive integers $d$ and $n$, let $[n]^d$ be the set of all vectors $(a_1,a_2,\dots, a_d)$, where $a_i$ is an integer with $0\leq a_i\leq n-1$. A subset $S$ of $[n]^d$ is called a \emph{Sidon set} if all sums of two (not necessarily…

A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$…

Let n be a non-null positive integer and $d(n)$ is the number of positive divisors of n, called the divisor function. Of course, $d(n) \leq n$. $d(n) = 1$ if and only if $n = 1$. For $n > 2$ we have $d(n) \geq 2$ and in this paper we try to…

A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel…

We prove that at differentiability points $r_0>0$ of the volume function of a compact set $A\subset {\mathbb R}^d$ (associating to $r$ the volume of the $r$-parallel set of $A$), the surface area measures of $r$-parallel sets of $A$…

A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\ldots,n\}$ of density at least…

Let $n\geq 1$, $0\leq t\leq {n \choose 2}$ be arbitrary integers. Define the numbers $I_n(t)$ as the number of permutations of $[n]$ with $t$ inversions. Let $n,d\geq 1$ and $0\leq t\leq (d-1)n$ be arbitrary integers. Define {\em the…

Erd\H{o}s and Hall defined a pair $(m, n)$ of positive integers to be interlocking, if between any pair of consecutive divisors (both larger than $1$) of $n$ (resp. $m$) there is a divisor of $m$ (resp. $n$). A positive integer is said to…

For a point set of $n$ elements in the $d$-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers $n$, $d$ and under the assumption of a…

The integer $d=\prod_{i=1}^s p_i^{b_i}$ is called an exponential divisor of $n=\prod_{i=1}^s p_i^{a_i}>1$ if $b_i \mid a_i$ for every $i\in \{1,2,...,s\}$. Let $\tau^{(e)}(n)$ denote the number of exponential divisors of $n$, where…

Finding a good bound on the maximal edge diameter $\Delta(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known,…

A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q…

Let $F_n$ be a free group of rank $n$, with free generating set $X$. A subset $D$ of $F_n$ is a \emph{Distinct Difference Configuration} if the differences $g^{-1}h$ are distinct, where $g$ and $h$ range over all (ordered) pairs of distinct…

In this paper, we prove Erd\H{o}s distance conjecture in $\mathbb{R}^d$, namely, a set of $n$ points in $\mathbb{R}^2$ determines $\Omega(\frac{n}{\sqrt{\log n}})$ distances, and for $d\ge 3$, a set of $n$ points in $\mathbb{R}^d$…

Suppose $(M^{n},g)$ is a Riemannian manifold with nonnegative Ricci curvature, and let $h_{d}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $d$. Colding and Minicozzi proved that…

Let $A\subseteq [N]$ be such that for any pair of distinct subsets $B,C\subset A$, the products $\prod_{b\in B}b$ and $\prod_{c\in C}c$ are distinct. We prove that $|A|\leq \pi(N)+\pi(N^{1/2})+o(\pi(N^{1/2}))$, where $\pi$ is the prime…

Given integer $n > 0$ and $m > 1$, we call a partition of set $[n] = \{1, \dots, n\}$ {\em $m$-good} if each of the partitioning sets is of size at most $m$ and the sum of numbers in it is a power of $m$, that is, $m^t$ for some $t \geq 0$.…

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection $\B\subset 2^{[n]}$ forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps…

Erd\H{o}s and Graham proposed to determine the number of subsets $S \subseteq \left\{1,2,\dots,n\right\}$ with $\sum_{s \in S} 1/s = 1$ and asked, among other things, whether that number could be as large as $2^{n - o(n)}$. We show that the…

Burkill and Mirsky, and Kalmanson, prove independently that, for every $r\ge 2, n\ge 1$, there is a sequence of $r^{2^n}$ vectors in $\mathbb R^n$, which does not contain a subsequence of $r+1$ vectors $v^1, v^2,\dots,v^{r+1}$ such that,…