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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 $0\le a\le b\le 1$, there exists an infinite
sequence of positive integers $A$ with lower density $a$ and upper
density $b$ such that the sets $A$ and $\text{N}\setminus A$ contain no
infinite arithmetic and geometric progressions. Furthermore, for
any $m\ge 2$ and any positive numbers $a_1,\dots ,a_m$
satisfying $a_1+\cdots +a_m=1,$ we give an explicit partition of
$\text{N}$ 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 $\text{N}$ 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

https://hrcak.srce.hr/23567

Datum izdavanja:

28.5.2008.

Posjeta: 2.036 *