Izvorni znanstveni članak
Partitions of positive integers into sets without infinite progressions
Artūras Dubickas
orcid.org/0000-0002-3625-9466
; Department of Mathematics and Informatics, Vilnius University, Lithuania
Sažetak
We prove a result which implies that, for any real numbers $a$ and
$b$ satisfying $0 \leq a \leq b \leq 1$, there exists an infinite
sequence of positive integers $A$ with lower density $a$ and upper
density $b$ such that the sets $A$ and $\N \setminus A$ contain no
infinite arithmetic and geometric progressions. Furthermore, for
any $m \geq 2$ and any positive numbers $a_1, \dots, a_m$
satisfying $a_1+\dots+a_m=1,$ we give an explicit partition of
$\N$ into $m$ disjoint sets $\cup_{j=1}^m A_j$ such that
$d_{P}(A_j)=a_j$ for each $j=1,\dots,m$ and each infinite
arithmetic and geometric progression $P,$ where $d_{P}(A_j)$
denotes the proportion between the elements of $P$ that belong to
$A_j$ and all elements of $P,$ if a corresponding limit exists. In
particular, for $a=1/2$ and $m=2,$ this gives an explicit
partition of $\N$ into two disjoint sets such that half of
elements in each infinite arithmetic and geometric progression
will be in one set and half in another.
Ključne riječi
infinite sequence; partition of integers; density; arithmetic and geometric progression
Hrčak ID:
23567
URI
Datum izdavanja:
28.5.2008.
Posjeta: 1.570 *