"A sea-side town where every house can see the sea"

2011-04-23T01:16:08Z

This is a reference request.

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):

There exists a set $S$, which is a subset of the unit square $[0,1]^2\subset\mathbb{R}^2$ with full Lebesgue measure, with the property that for any point $p\in S$ there exists a line $\ell \subset \mathbb{R}^2$ such that $\ell \cap S = { p}$.

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).

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.)

Answer by Mark Lewko for "A sea-side town where every house can see the sea"

2011-04-23T01:30:02Z

This is a Nikodymn set. I haven't seen a citation to Nikodymn's original paper, but the history is breifly discussed (with references) in Stein's survey article Singular Integrals: The Roles of Calderón and Zygmund.

Edit: Stein's Harmonic Analysis cites this to: "O. Nikodym, Sur les ensembles accessibles, Fund. Math. 1927 10:116-168"

"A sea-side town where every house can see the sea" - MathOverflow

"A sea-side town where every house can see the sea"

Answer by Mark Lewko for "A sea-side town where every house can see the sea"