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Statement

Measure universality is as on the page of Theorem 1.3 (p. 1).

Theorem 1.5 (Kolountzakis; p. 3). Let $A\subset\mathbb R$ be an infinite set that contains, for each $n\in\mathbb N$, elements $a_1>a_2>\cdots>a_n>0$ such that $-\log(\delta_n)=o(n)$, where

$$\delta_n=\min_{i\in\{1,\ldots,n-1\}}\frac{a_i-a_{i+1}}{a_1}.$$

Then $A$ is not measure universal.

The elements $a_1,\ldots,a_n$ may depend on $n$. The survey reads the theorem as saying that a set containing arbitrarily long chunks of slowly decaying sequences is not measure universal, notes that it generalizes Theorem 1.3, and records that it shows the sumsets $\{2^{-n^\alpha}\}+\{2^{-n^\alpha}\}$ are not measure universal for every $\alpha\in(0,2)$ (p. 3).

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 was read clause by clause on p. 3. The survey quotes the theorem from Kolountzakis's paper of 1997, which was not read here.

Proof pointer

No proof in the survey. It records (p. 3) that Kolountzakis's method is probabilistic: the avoiding set is built by choosing basic intervals independently at random.

Dependencies

M. N. Kolountzakis, Infinite patterns that can be avoided by measure, Bull. London Math. Soc. 29 (1997), no. 4, 415--424.

Bears on

• Problem 120: answers the question affirmatively for every infinite set meeting the chunk condition, which covers the sublacunary case of Theorem 1.3. For the set $\{2^{-k}\}$ every choice of $n$ elements has $\delta_n<2^{2-n}$, so $-\log(\delta_n)$ is not $o(n)$ and the theorem does not apply to it; the problem is not settled.