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Statement

Measure universality is as on the page of Theorem 1.3: every measurable set of positive Lebesgue measure contains some $\lambda A+t$ with $\lambda\ne0$ (p. 1).

Theorem 1.4 (Bourgain; p. 2). If $A_1,A_2,A_3\subseteq\mathbb R$ are infinite, then the sumset

$$A_1+A_2+A_3=\{a+b+c:a\in A_1,\ b\in A_2,\ c\in A_3\}$$

is not measure universal.

The survey notes that this cannot be deduced from Theorem 1.3, because the sets $A_i$ may be sequences of arbitrarily rapid decay (p. 2).

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. 2. The survey quotes the theorem from Bourgain's paper, which was not read here.

Proof pointer

No proof in the survey. It records (p. 2) that Bourgain characterized measure universal sets $X$ by an integral inequality over finite subsets of $X$ and continuous functions on tori (its display (1.2)), and proved the theorem by building a random function, a sum of indicators of small cubes, that violates that inequality when three infinite sets add; it refers to an exposition of the proof by T. Tao.

Dependencies

J. Bourgain, Construction of sets of positive measure not containing an affine image of a given infinite structure, Israel J. Math. 60 (1987), no. 3, 333--344.

Bears on

• Problem 120: answers the question affirmatively for every infinite set containing a sumset $A_1+A_2+A_3$ of three infinite sets of reals, including sums of rapidly decaying sequences. It does not decide a single sequence such as $2^{-n}$, which the survey records as open (p. 2), and does not settle the problem.