Related papers: On Waring's problem: beyond Freiman's theorem

Suppose that $\{a_j\}\in \ell^1$, and suppose that for any sequence $(t_n)$ of integers there exits a constant $C_1>0$ such that $$\sharp\left\{k\in\mathbb{Z}:\sup_{n\geq 1}\left|\sum_{i\in \mathcal{B}_n-t_n}…

An order-preserving Freiman 2-isomorphism is a map $\phi:X \rightarrow \mathbb{R}$ such that $\phi(a) < \phi(b)$ if and only if $a < b$ and $\phi(a)+\phi(b) = \phi(c)+\phi(d)$ if and only if $a+b=c+d$ for any $a,b,c,d \in X$. We show that…

In 1971 Cusick proved that every real number $x\in[0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$ S(k):= \{ x\in[0,1] : a_n(x) \geq k \text{ for all }…

Let $\mathcal{B} = (B_1,\ldots, B_h)$ be an $h$-tuple of sets of positive integers. Let $g_{\mathcal{B} }(n)$ count the number of representations of $n$ in the form $n = b_1\cdots b_h$, where $b_i \in B_i$ for all $i \in \{1,\ldots, h\}$.…

We prove that for all $n\geq 1$ there exists a number between $n^2$ and $(n+1)^2$ with at most 4 prime factors. This is the first result of this kind that holds for every $n\geq 1$ rather than just sufficiently large $n$. Our approach…

In his book "250 Problems in Elementary Number Theory", W.Sierpinski shows that the numbers 1+2^(2^n)+2^(2^n+1) are divisible by 21; for n=1,2,.... In this paper, we prove a similar but more general result.Consider the natural numbers of…

Let g_j denote the largest integer that is represented exactly j times as a non-negative integer linear combination of { x_1, ... , x_n. We show that for any k > 0, and n = 5, the quantity g_0 - g_k is unbounded. Furthermore, we provide…

In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

We prove that there are infinitely many integers $n$ such that $n$ and $n+1$ have the same number of distinct prime divisors.

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions of the equation $n=s+s'$, $s, s'\in S$, $s<s'$. In this paper, we determine the…

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph…

The Frobenius number of relatively prime positive integers $a_1, \ldots, a_n$ is the largest integer that is not a nononegative integer combination of the $a_i.$ Given positive integers $a_1, \ldots, a_n$ with $n \ge 2,$ the set of…

We show that for every $0 < \epsilon \leq 1$ and integer $k\geq 1$, there exists an integer $n = n(\epsilon,k)$ so that for all primes $p$, and integers $0 \leq a \leq p-1$, there exist integers $1 \leq x_1 < ... < x_n \leq p^\epsilon$ such…

Given a bounded measurable function $\sigma$ on $\mathbb{R}^n$, we let $T_\sigma $ be the operator obtained by multiplication on the Fourier transform by $\sigma $. Let $0<s_1\le s_2\le \cdots \le s_n<1$ and $\psi$ be a Schwartz function on…

Choose a polynomial $f$ uniformly at random from the set of all monic polynomials of degree $n$ with integer coefficients in the box $[-L,L]^n$. The main result of the paper asserts that if $L=L(n)$ grows to infinity, then the Galois group…

S\'ark\"ozy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of S\'ark\"ozy's theorem…

We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $\Omega(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends…

In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic…