Construction techniques for some thin sets in duals of compact abelian groups

Hajela, D. J.. "Construction techniques for some thin sets in duals of compact abelian groups." Annales de l'institut Fourier 36.3 (1986): 137-166. <http://eudml.org/doc/74721>.

@article{Hajela1986,
abstract = {Various techniques are presented for constructing $\Lambda $ (p) sets which are not $\Lambda (p+\epsilon )$ for all $\epsilon &gt;0$. The main result is that there is a $\Lambda $ (4) set in the dual of any compact abelian group which is not $\Lambda (4+\epsilon )$ for all $\epsilon &gt;0$. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $\Lambda $ (p) not $\Lambda (p+\epsilon )$ sets, for certain values of $p$. The main new constructions in specific dual groups are:– there is a $\Lambda $ (2k) set which is not $\Lambda (2k+\varepsilon )$ in $\{\bf Z\}(2)\oplus \{\bf Z\}(2)\oplus \cdots $ for all $2\le k$, $k\in \{\bf N\}$ and $\varepsilon &gt;0$, and in $\{\bf Z\}(p)\oplus \{\bf Z\}(p)\oplus \cdots $ ($p$ a prime, $p&gt;2$) for $2\le k&lt; p$, $k\in \{\bf N\}$ and $\varepsilon &gt;0$ (answering a question in J. Lopez and K. Ross, Marcel Dekker, 1975),– there is a $\Lambda $ (2k) set which is not $\Lambda (4k-4+\varepsilon )$ in $\{\bf Z\}(p^\{\infty \})$ for $2\le k$, $k\in \{\bf N\}$ and all $\epsilon &gt;0.$It is also shown that random infinite integer sequences are $\Lambda $ (2k) but not $\Lambda (2k+\epsilon )$ for $2\le k$, $k\in \{\bf N\}$ and $\epsilon &gt;0$.},
author = {Hajela, D. J.},
journal = {Annales de l'institut Fourier},
keywords = {thin sets; duals of compact abelian groups; random infinite integer sequences},
language = {eng},
number = {3},
pages = {137-166},
publisher = {Association des Annales de l'Institut Fourier},
title = {Construction techniques for some thin sets in duals of compact abelian groups},
url = {http://eudml.org/doc/74721},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Hajela, D. J.
TI - Construction techniques for some thin sets in duals of compact abelian groups
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 3
SP - 137
EP - 166
AB - Various techniques are presented for constructing $\Lambda $ (p) sets which are not $\Lambda (p+\epsilon )$ for all $\epsilon &gt;0$. The main result is that there is a $\Lambda $ (4) set in the dual of any compact abelian group which is not $\Lambda (4+\epsilon )$ for all $\epsilon &gt;0$. Along the way to proving this, new constructions are given in dual groups in which constructions were already known of $\Lambda $ (p) not $\Lambda (p+\epsilon )$ sets, for certain values of $p$. The main new constructions in specific dual groups are:– there is a $\Lambda $ (2k) set which is not $\Lambda (2k+\varepsilon )$ in ${\bf Z}(2)\oplus {\bf Z}(2)\oplus \cdots $ for all $2\le k$, $k\in {\bf N}$ and $\varepsilon &gt;0$, and in ${\bf Z}(p)\oplus {\bf Z}(p)\oplus \cdots $ ($p$ a prime, $p&gt;2$) for $2\le k&lt; p$, $k\in {\bf N}$ and $\varepsilon &gt;0$ (answering a question in J. Lopez and K. Ross, Marcel Dekker, 1975),– there is a $\Lambda $ (2k) set which is not $\Lambda (4k-4+\varepsilon )$ in ${\bf Z}(p^{\infty })$ for $2\le k$, $k\in {\bf N}$ and all $\epsilon &gt;0.$It is also shown that random infinite integer sequences are $\Lambda $ (2k) but not $\Lambda (2k+\epsilon )$ for $2\le k$, $k\in {\bf N}$ and $\epsilon &gt;0$.
LA - eng
KW - thin sets; duals of compact abelian groups; random infinite integer sequences
UR - http://eudml.org/doc/74721
ER -