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Statement

Question (printed p. 138, quoted). "I could not decide the following question. Suppose $S$ is a measurable set of positive reals such that the quotient of two distinct elements of $S$ is never an integer. Is $\mu(S)$ necessarily finite?"

Footnote 2 (p. 138, quoted). "Added in proof P. ERDŐS informed me that E. SZEMERÉDI can show that $\mu(S)$ is not necessarily finite."

The paper gives no construction or reference for Szemerédi's answer.

Source. W. M. Schmidt, Disproof of some conjectures on Diophantine approximations, Studia Sci. Math. Hungar. 4 (1969), 137--144; the question and its footnote 2 on printed p. 138, read on the page image. The edition read is identified on the source card.

Read depth. Claims checked: the question and footnote 2 were read clause by clause on the page image of p. 138. Nothing here is independently reviewed.

Proof pointer

None; this is a question, with a reported answer that the paper does not prove.

Dependencies

None.

Bears on

• Problem 1195: the problem takes a set of infinite measure in which no ratio of two distinct elements is an integer and asks how fast its measure in $(0,x)$ can grow; Schmidt's question is whether a set of positive reals with this property can have infinite measure at all, and his footnote records only Erdős's report that Szemerédi can show it can. The page bears on the problem's background and says nothing about the growth rate it asks for.