"A sea-side town where every house can see the sea" - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:33:45Z http://mathoverflow.net/feeds/question/62689 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62689/a-sea-side-town-where-every-house-can-see-the-sea "A sea-side town where every house can see the sea" Willie Wong 2011-04-23T01:16:08Z 2011-04-23T01:36:46Z <p>This is a reference request. </p> <p>The phrase in the title is, if I remember correctly, how Eli Stein described the following set (the definition may be faulty, but I think it is right): </p> <blockquote> <p>There exists a set $S$, which is a subset of the unit square $[0,1]^2\subset\mathbb{R}^2$ with <strong>full Lebesgue measure</strong>, with the property that for any point $p\in S$ there exists a line $\ell \subset \mathbb{R}^2$ such that $\ell \cap S = { p}$. </p> </blockquote> <p>I vaguely remember the theorem being attributed to one of the Rieszes (not sure M. or F.; I might also be completely wrong about that). </p> <p>Can someone tell me whether my rough statement of the theorem above is correct, and better yet, can someone give a reference to the paper which proved it? (I know a paper exists; I remember digging it up in the bowels of the library when I was a beginning graduate student. I am sad to say that two moves later I can no longer find my photocopy of that article.)</p> http://mathoverflow.net/questions/62689/a-sea-side-town-where-every-house-can-see-the-sea/62691#62691 Answer by Mark Lewko for "A sea-side town where every house can see the sea" Mark Lewko 2011-04-23T01:30:02Z 2011-04-23T01:36:46Z <p>This is a <a href="http://en.wikipedia.org/wiki/Nikodym_set" rel="nofollow">Nikodymn set</a>. I haven't seen a citation to Nikodymn's original paper, but the history is breifly discussed (with references) in Stein's survey article <a href="http://www.ams.org/notices/199809/stein.pdf" rel="nofollow">Singular Integrals: The Roles of Calderón and Zygmund</a>.</p> <p>Edit: Stein's Harmonic Analysis cites this to: "O. Nikodym, Sur les ensembles accessibles, Fund. Math. 1927 10:116-168"</p>