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A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset of $A \cap S$ which intersects two opposite sides of the square $S$. Clearly, the 45 degree line $\{ (\pi + t, t) : t \in R \}$ does not cross any dyadic square. Does every non-trivial closed, connected set that is not a line segment cross some dyadic square?

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I would have expected the condition $m\le n$. Is that what you mean? Or is this another version of the problem? Every line segment that is not $\pm45$ degrees, and every sufficiently smooth curve that is not a $\pm45$ degree line segment must cross a dyadic square (even with the strong definition of dyadic square). So if there are counterexamples they are pretty pathological. –  Günter Rote Jan 31 '13 at 17:01
I think that every path-connected component of such a non-crossing set should be a line segment. –  Marco Golla Jan 31 '13 at 19:44
Gunter Rote, I do not mean the condition $m\leq n$. E.g. the square $[1/8,5/8]\times [1/4,3/4]$ is a dyadic square with my definition. Perhaps you would like to call such a square a translated dyadic square. –  Kevin Johnson Feb 7 '13 at 8:54
There used to be an interesting answer posted here. Where is it? Was it wrong? –  Günter Rote Feb 17 '13 at 13:17
Hi Gunter, I pointed out that there were two mistakes in this answer. A day later it was removed by somebody. –  Kevin Johnson Feb 18 '13 at 9:01

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