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'Behrend's construction'
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| Title of object: |
Behrend's construction |
| Canonical Name: |
BehrendsConstruction |
| Type: |
Result |
| Created on: |
2003-06-12 01:19:13 |
| Modified on: |
2003-06-12 12:08:45 |
| Classification: |
msc:11B25, msc:05D10 |
Revision comment (for changes between this and next version):
| Added information about Moser's constuction. |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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\renewcommand{\bibname}{References} |
Content:
At first sight it may seem that the greedy algorithm yields the densest subset of $\{0,1,\dotsc,N\}$ that is free of arithmetic progressions of length $3$. It is not hard to show that the greedy algorithm yields the set of numbers that lack digit $2$ in their ternary development. Density of such numbers is $O(N^{\log_3 2-1})$.
$N e^{-\sqrt{\ln N}(c\ln 2+2/c+o(1)}$. To maximize this value we can set $c=\sqrt{2/\ln 2}$. Thus, there exists a progression-free set of size at least
\begin{equation*}
N e^{-\sqrt{8\ln 2 \ln N}(1+o(1))}.
\end{equation*}
This result was later generalized to sets not containing arithmetic progressions of length $k$ by Rankin\cite{cite:rankin_behrgen}. His construction is more complicated, and depends on the estimates of the number of representations of an integer as a sum of many squares. He proves that the size of a set free of $k$-term arithmetic progression is at least
\begin{equation*}
N e^{-c (\log N)^{1/(k-1)}}.
\end{equation*}
\begin{thebibliography}{1}
\bibitem{cite:behrend_szem_low}
Felix~A. Behrend.
\newblock On the sets of integers which contain no three in arithmetic
progression.
\newblock {\em Proc. Nat. Acad. Sci.}, 23:331--332, 1946.
\newblock \PMlinkexternal{Zbl 0060.10302}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0060.10302}.
\bibitem{cite:rankin_behrgen}
Robert~A. Rankin.
\newblock Sets of integers containing not more than a given number of terms in
arithmetical progression.
\newblock {\em Proc. Roy. Soc. Edinburgh Sect. A}, 65:332--344, 1962.
\newblock \PMlinkexternal{Zbl 0104.03705}{http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0104.03705}.
\end{thebibliography}
%@ARTICLE{cite:behrend_szem_low,
% author = {Felix A. Behrend},
% title = {On the sets of integers which contain no three in arithmetic %progression},
% journal = {Proc. Nat. Acad. Sci.},
% volume = 23,
% year = 1946,
% pages = {331--332}
%} |
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