Related papers: A note on cube-free problems

By studying the n-dimensional cube, the author finds a new way, and use it to proves that the indefinite equation $x^n+y^n=z^n(n=>3)$ has no any solutions of positive integers.

Given a linear equation L, a set A of integers is L-free if A does not contain any non-trivial solutions to L. Meeks and Treglown showed that for certain kinds of linear equations, it is NP-complete to decide if a given set of integers…

We prove that for all squarefree $m$ and any set $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero squares the bound $|A|\leq m^{1/2}(3n)^{1.5n}$ holds, where $n$ denotes the number of odd prime divisors of $m$.

Cameron and Erd\H{o}s raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal…

A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least…

Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$…

We prove that the largest $Q_2$-free family of subsets of $[n]$ which contains sets of at most three different sizes has at most $(3 + 2\sqrt {3})N/3 + o(N) \approx 2.1547N + o(N)$ members, where $N = {n \choose {\lfloor n/2 \rfloor}}$.…

A nonzero rational number is called a cube sum if it is of form $a^3+b^3$ with $a,b\in \mathbb{Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$…

The $\mathcal{N}$ poset consists of four distinct sets $W,X,Y,Z$ such that $W\subset X$, $Y\subset X$, and $Y\subset Z$ where $W$ is not necessarily a subset of $Z$. A family $\mathcal{F}$ as a subposet of the $n$-dimensional Boolean…

Let $d$ be a fixed positive integer and let $\epsilon>0$. It is shown that for every sufficiently large $n\geq n_0(d,\epsilon)$, the $d$-dimensional unit cube can be decomposed into exactly $n$ smaller cubes such that the ratio of the side…

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as…

Let $\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j]…

We show that for all $A, B \subseteq \{0,1,2\}^{d}$ we have $$ |A+B|\geq (|A||B|)^{\log(5)/(2\log(3))}. $$ We also show that for all finite $A,B \subset \mathbb{Z}^{d}$, and any $V \subseteq\{0,1\}^{d}$ the inequality $$ |A+B+V|\geq…

We study the maximum size of a subset of the $n \times n$ integer grid that does not contain specific geometric configurations, a variation of the classical problems initiated by Erd\H{o}s and Purdy. While extremal problems for 3-point…

W.M.Schmit[11] conjectured that for any$\;\theta$ with deg$\;\theta\geq 3,$ there is no constant$\;C=C(\theta)$ so that$\;|p-q\theta|>Cq^{-1}$ for every rationa$\;p/q.$ [12,p26] states that the computations of the first several thousand…

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that $d \leq (2n-3)/5$ for every connected $C_4$-free graph of order $n$, diameter $d$, and edge-connectivity at least $3$, which is best possible up to a small…

For integers $m$ and $n$, we study the problem of finding good lower bounds for the size of progression-free sets in $(\mathbb{Z}_{m}^{n},+)$. Let $r_{k}(\mathbb{Z}_{m}^{n})$ denote the maximal size of a subset of $\mathbb{Z}_{m}^{n}$…

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…

Feigin and Shoikhet conjectured in math/0610410 that successive quotients $B_m(A_n)$ of the lower central series filtration of a free associative algebra $A_n$ have polynomial growth. In this paper we give a proof of this conjecture, using…

We prove that any set of points in $\mathbb{R}^d$, any three of which form an angle less than $\frac{\pi}{3} + c$, has size $(1+\Theta(c))^d$ for sufficiently small $c>0$. The proof is based on a refinement of an approach by Erd\H{o}s and…