Zobrazeno 1 - 10
of 348
pro vyhledávání: '"Covert, David"'
We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-S\'{a}rk\"{o}zy theorem on squares in sets of integers w
Externí odkaz:
http://arxiv.org/abs/1808.06665
Let $\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb F_q$ with $q$ elements. Given $k$ sets $E_j\subset \mathbb F_q^d$ for $j=1,2,\ldots, k$, the generalized $k$-resultant modulus set, denoted by $\Delta_k(E_1,E_2, \
Externí odkaz:
http://arxiv.org/abs/1703.00609
We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution fo
Externí odkaz:
http://arxiv.org/abs/1609.02090
Autor:
Covert, David, Senger, Steven
We obtain bounds on the number of triples that determine a given pair of dot products arising in a vector space over a finite field or a module over the set of integers modulo a power of a prime. More precisely, given $E\subset \mathbb F_q^d$ or $\ma
Externí odkaz:
http://arxiv.org/abs/1508.02691
We study the $k$-resultant modulus set problem in the $d$-dimensional vector space $\mathbb F_q^d$ over the finite field $\mathbb F_q$ with $q$ elements. Given $E\subset \mathbb F_q^d$ and an integer $k\ge 2$, the $k$-resultant modulus set, denoted b
Externí odkaz:
http://arxiv.org/abs/1508.02688
For a set $E\subset \mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \Delta_k(E) =\{\|\textbf{x}_1 + \dots + \textbf{x}_k\|\in \mathbb F_q: \textbf{x}_1, \dots, \textbf{x}_k \in E\},$ where $\|\textbf{v}\|=v_1^2+\cdots+ v_d^2$ for $\tex
Externí odkaz:
http://arxiv.org/abs/1403.6138
Autor:
Covert, David
The Erd\H os-Falconer distance problem in $\mathbb{Z}_q^d$ asks one to show that if $E \subset \mathbb{Z}_q^d$ is of sufficiently large cardinality, then $\Delta(E) := \{(x_1 - y_1)^2 + \dots + (x_d - y_d)^2 : x, y \in E\}$ satisfies $\Delta(E) = \ma
Externí odkaz:
http://arxiv.org/abs/1309.1495
Autor:
Covert, David
Given $E \subset \mathbb{F}_q^d$, we show that certain configurations occur frequently when $E$ is of sufficiently large cardinality. Specifically, we show that we achieve the statistically number of $k$-stars $\displaystyle\left|\left\{(x, x^1, \dot
Externí odkaz:
http://arxiv.org/abs/1309.1497
We study variants of the Erd\H os distance problem and dot products problem in the setting of the integers modulo $q$, where $q = p^{\ell}$ is a power of an odd prime.
Comment: 21 pages
Comment: 21 pages
Externí odkaz:
http://arxiv.org/abs/1105.5373
Autor:
Banks, William D., Covert, David
We estimate the sizes of the sumset A + A and the productset A $\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y.
Comment: 12 pages
Comment: 12 pages
Externí odkaz:
http://arxiv.org/abs/1010.3322