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Source. The conjecture on p. 123 of P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978/79), no. 2, 113--138, doi:10.2307/44151159, as identified on the source card. Pages are those of the journal print; the conjecture is unnumbered.

Read depth. Claims checked: the passage was read clause by clause on p. 123.

Statement

The finite case (p. 123). If $S$ is a set of positive measure on the line and $A$ a finite subset of the line, then $S$ contains a set similar to $A$. The paper derives this from the Lebesgue density theorem, says it is substantially due to Steinhaus and often rediscovered, and explains "similar" as containing "a set $A'$ which can be transformed into $A$ by a fractional linear transformation" [sic]. The density argument gives copies under the similarities $x\mapsto\lambda x+\mu$ with $\lambda\ne0$, and the problem's statement below uses these maps.

The conjecture (p. 123, quoted). "I have conjectured for a long time that if $A$ is any infinite set on the line then there always is a subset $S$ of the line of positive measure which does not contain a set similar to $A$."

The paper notes that one may assume without loss of generality that $A$ is a sequence of positive numbers tending to $0$. If the conjecture holds, it asks further: for a countable set $A$ of $[0,1]$, determine or estimate the largest possible measure of a subset $S$ of $[0,1]$ that contains no set similar to $A$.

Proof pointer

None for the conjecture. The finite case follows from the density theorem, as the paper states without detail.

Dependencies

None.

Bears on

• Problem 120: the conjecture is the problem, whose site statement asks for a set of positive measure containing no set $aA+b$ with $a\ne0$. The paper states it as open and proves nothing toward it.