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Statement

A nontrivial affine copy of $A\subset\mathbb R$ is $\lambda A+t$ with $\lambda\ne0$ and $t\in\mathbb R$; $A$ is measure universal when every measurable subset of $\mathbb R$ of positive Lebesgue measure contains a nontrivial affine copy of $A$ (p. 1). A decreasing sequence with $\lim_{n\to\infty}a_{n+1}/a_n=1$ is called sublacunary (p. 2).

Theorem 1.3 (Eigen, Falconer; p. 2). Let $a_n\to0$ be a decreasing sequence. If

$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}=1,$$

then $(a_n)_{n=1}^\infty$ is not measure universal.

The survey records that Falconer (Proc. Amer. Math. Soc. 90 (1984), 77--78) and Eigen (Studia Sci. Math. Hungar. 20 (1985), 411--412) proved this independently by a direct Cantor set construction, and that no faster-decreasing sequence is known not to be measure universal, the sequence $(2^{-n})$ being the main open case at the time of writing (p. 2). Since a superset of a set that is not measure universal is not measure universal (p. 2), every set containing such a sequence is not measure universal either.

Source. Yeonwook Jung, Chun-Kit Lai and Yuveshen Mooroogen, Fifty years of the Erdős similarity conjecture, arXiv:2412.11062v2 (1 January 2025), whose labels and page numbers are cited here; the edition is identified on the source card.

Read depth. Claims checked: the statement and definitions were read clause by clause on pp. 1--2. The survey quotes the theorem from the papers of Eigen and of Falconer, which were not read here.

Proof pointer

The survey gives no proof at this label. Its Theorem 2.1(1), proved on pp. 5--7, gives a new proof: a strictly decreasing sublacunary sequence is not even bi-Lipschitz measure universal, and an affine map with $\lambda\ne0$ is bi-Lipschitz; see Theorem 2.1. The survey also notes that Kolountzakis's Theorem 1.5 generalizes this theorem (p. 3).

Dependencies

K. J. Falconer, On a problem of Erdős on sequences and measurable sets, Proc. Amer. Math. Soc. 90 (1984), no. 1, 77--78; S. J. Eigen, Putting convergent sequences into measurable sets, Studia Sci. Math. Hungar. 20 (1985), 411--412.

Bears on

• Problem 120: answers the question affirmatively for every infinite set containing a decreasing sequence $a_n\to0$ with $a_{n+1}/a_n\to1$: some set of positive measure contains no nontrivial affine copy of it. It says nothing about sequences that decrease geometrically or faster, such as $2^{-n}$, and does not settle the problem.