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Statement

Here $m$ is Lebesgue measure.

Theorem 1.6 (Kolountzakis; p. 3). For every infinite set $A\subseteq\mathbb R$:

1. there is a measurable set $E_1\subset[0,1]$, of Lebesgue measure as close to $1$ as desired, such that

$$m\bigl(\{y:\ x+yA\subset E_1\text{ for some }x\in\mathbb R\}\bigr)=0;$$

2. there is a measurable set $E_2$ of positive Lebesgue measure such that the set $\{(\lambda,t)\in\mathbb R^2:\lambda A+t\subset E_2\}$ has two-dimensional Lebesgue measure zero.

The survey calls this an almost everywhere solution to the Erdős similarity problem (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; the method is Kolountzakis's probabilistic construction (p. 3). The survey's Theorem 6.1 (p. 19) is the analogue for the variant in the large, with almost every dilation $y$ (p. 20).

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: an almost-everywhere form of the question for every infinite set. The sets $E_1$ and $E_2$ may still contain affine copies of $A$, for a null set of dilations $y$ in the case of $E_1$ and a planar null set of pairs $(\lambda,t)$ in the case of $E_2$, so the theorem does not answer the problem for any particular set and does not settle it.