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Angel Cruz, Chun-Kit Lai and Malabika Pramanik, A proof of the Erdös similarity conjecture, arXiv:2001.02395v1, submitted 8 January 2020. The abstract claims that for every infinite set $A\subseteq\mathbb{R}$ there is a compact set $E\subseteq\mathbb{R}$ of positive Lebesgue measure containing no nontrivial affine copy $aA+b$, $a\ne0$, of $A$, which is the question of Problem 120 answered yes for every infinite $A$.

Withdrawal. Version 2, of 11 January 2020, withdraws the paper. Its comment says that a gap was found in Proposition 3.3 and thanks Vjekoslav Kovač for pointing it out; the record offers no v2 text and no repaired argument. The claim therefore stands as withdrawn, not refuted, and the problem's standing does not rest on it. The later survey of Jung, Lai and Mooroogen, carded as [[../library/analysis/jung_2024_fifty_years_erdos_similarity_conjecture/_index|Jung, Lai and Mooroogen 2024]], records the conjecture as open.