Set of real numbers with positive measure containing no midpoints - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:05:52Z http://mathoverflow.net/feeds/question/32117 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32117/set-of-real-numbers-with-positive-measure-containing-no-midpoints Set of real numbers with positive measure containing no midpoints Lieven 2010-07-16T06:13:30Z 2010-07-16T12:19:03Z <p>Does there exists a subset E of R with positive measure and without containing any midpoints (i.e. x,y distinct in E, (x+y)/2 not in E)?</p> http://mathoverflow.net/questions/32117/set-of-real-numbers-with-positive-measure-containing-no-midpoints/32124#32124 Answer by Gerry Myerson for Set of real numbers with positive measure containing no midpoints Gerry Myerson 2010-07-16T06:54:23Z 2010-07-16T06:54:23Z <p>According to James Foran, Non-averaging sets, dimension, and porosity, Canad Math Bull 29 (1986) 60-63, "It follows from the Lebesgue Density Theorem that a measurable, non-averaging subset (of $(0,1]$) cannot have positive measure." </p> http://mathoverflow.net/questions/32117/set-of-real-numbers-with-positive-measure-containing-no-midpoints/32144#32144 Answer by Benoît Kloeckner for Set of real numbers with positive measure containing no midpoints Benoît Kloeckner 2010-07-16T10:42:16Z 2010-07-16T10:42:16Z <p>As already said by Gerry, the answer to your question is negative. However, it becomes positive if you only ask your set to have Hausdorff dimension 1 instead of positive Lebesgue measure, see </p> <p>Salem, R.; Spencer, D. C. <em>On sets which do not contain a given number of terms in arithmetical progression.</em> Nieuw Arch. Wiskunde (2) 23, (1950). 133--143.</p> <p>For a more general recent result see</p> <p>Tamás Keleti <em>Construction of one-dimensional subsets of the reals not containing similar copies of given patterns</em>, Analysis and PDE Vol. 1 (2008), No. 1, 29-33 </p> <p>(if you do not know this journal, you should have a look at it and more generally to the web site of the <em>Mathematical Science publishers</em>, by the way.)</p> http://mathoverflow.net/questions/32117/set-of-real-numbers-with-positive-measure-containing-no-midpoints/32146#32146 Answer by Vagabond for Set of real numbers with positive measure containing no midpoints Vagabond 2010-07-16T10:43:30Z 2010-07-16T11:27:28Z <p>No, such a set cannot exist and one can prove this using Lebesgue Density Theorem and a simple pegionhole argument. Infact all points $x$ which are density points of $E$ will be a midpoint for some $y,z \in E$ i.e., $x=\frac{y+z}{2}$. </p> <p>Let $F \subseteq E$ be the set of density points of E, and $x \in F$.</p> <p>Then there exists a $\epsilon > 0$ such that $m( B_{\epsilon}(x)\cap F) > \epsilon$. Now if $x$ is not a midpoint of $E$ then $\forall d \in (0,\epsilon)$, atleast one of $x-d$ or $x+d$ does not belong to $F$. </p> <p>But then $m( B_{\epsilon}(x)\cap F)= \int_0^{\epsilon} |F\cap \lbrace x-t,x+t\rbrace| dt &lt; \epsilon$, a contradiction !!</p> <p>A set $A$ of real number is called Universal if every measurable set of positive measure necessarily contains an affine image of $A$. A simple variation of the above argument will give that all finite set $A$ are infact Universal. However, no example of an infinite Universal set is knwon and its a conjecture of Erdos that no infinite universal sets exists.</p> <p>This paper has a nice discussion and references to this problem </p> <p>M. Kolountzakis: Infinite Patterns That Can Be Avoided by Measure, Bull. London Math. Soc. 29 (1997), 4, 415-424. <a href="http://fourier.math.uoc.gr/~mk/ps/universal.pdf" rel="nofollow">http://fourier.math.uoc.gr/~mk/ps/universal.pdf</a></p> <p>As Gerry and Benoît Kloeckner has mentioned the problem becomes interesting when one considers Hausdroff measure instead of Lebesgue measure. </p> <p>Recently I. Laba and M. Pramanik proved existence of 3 term arithmetic progression even in closed sets which has Hausdroff dimension close to 1, `under the condition that E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions'</p> <p>I. Laba and M. Pramanik: "Arithmetic progressions in sets of fractional dimension",, Geom. Funct. Anal. 19 (2009), 429-456. <a href="http://www.math.ubc.ca/~ilaba/preprints/progressions-may15.pdf" rel="nofollow">http://www.math.ubc.ca/~ilaba/preprints/progressions-may15.pdf</a></p>