Jung 2024 fifty years erdos similarity conjecture

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corollary_3_8: States the survey's corollary, drawn from a theorem of Shmerkin and Suomala on spatially independent martingales, that a Cantor set in the reals of positive Hausdorff dimension is not measure universal.

theorem_1_3: States the theorem of Eigen and of Falconer, as the survey gives it, that a decreasing sequence a_n tending to 0 with a_{n+1}/a_n tending to 1 is not measure universal, the slow-decay case of the Erdős similarity conjecture.

theorem_1_4: States Bourgain's theorem, as the survey gives it, that if A_1, A_2, A_3 are infinite subsets of the reals then the sumset A_1 + A_2 + A_3 is not measure universal.

theorem_1_5: States Kolountzakis's criterion, as the survey gives it: an infinite set of reals that contains, for each n, elements a_1 > ... > a_n > 0 whose minimal relative gap delta_n satisfies -log(delta_n) = o(n) is not measure universal.

theorem_1_6: States Kolountzakis's almost-everywhere result, as the survey gives it: for every infinite set A of reals there is a set in [0,1] of measure arbitrarily close to 1 containing x + yA for only a null set of y, and a set of positive measure containing lambda A + t for only a planar null set of (lambda, t).

theorem_2_1: States the Feng-Lai-Xiong theorem, as the survey gives it: a strictly decreasing sequence tending to 0 with a_{n+1}/a_n tending to 1 is not bi-Lipschitz measure universal, while one with limsup a_{n+1}/a_n < 1 maps bi-Lipschitz into every set of positive measure.

theorem_3_6: States the Gallagher-Lai-Weber theorem, as the survey gives it, that a Cantor set in the reals of positive Newhouse thickness is not full measure universal, and hence not measure universal.

theorem_4_2: States the theorem of Jung and Lai that for every Cantor set K in the reals there are a Cantor set K' and delta > 0 with K meeting lambda K' + t for all lambda in (1/(1+delta), 1+delta) and t in (-delta, delta), so that K is not topologically universal.

theorem_7_1: States the theorem of Burgin, Goldberg, Keleti, MacMahon and Wang, as the survey gives it: some measurable set in [0,1] of positive measure, with 0 as a Lebesgue density point, contains no sequence y b^{-n} with y nonzero and b > 1.

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Yeonwook Jung, Chun-Kit Lai, Yuveshen Mooroogen, Fifty years of the Erdős similarity conjecture. arXiv:2412.11062 (2024). The edition read is arXiv:2412.11062v2 of 1 January 2025 (stamp "arXiv:2412.11062v2 [math.CA] 1 Jan 2025" on p. 1), 26 pages whose printed page numbers equal the PDF page numbers; labels and pages below are that edition's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2412.11062), every other right reserved.

This is a survey, written fifty years after Erdős asked in 1974 whether any infinite set A in the reals is measure universal, meaning every positive-measure measurable set contains a nontrivial affine copy of A. It reviews the early results: Eigen and Falconer independently showed sublacunary sequences (a_{n+1}/a_n tending to 1) are not measure universal (Theorem 1.3), Bourgain's characterization and his theorem that a sum of three infinite sets is never measure universal (Theorem 1.4), and Kolountzakis's results (Theorems 1.5, 1.6). It then surveys three newer variants studied by the authors and collaborators, together with the uncountable case: the bi-Lipschitz variant, with the Feng-Lai-Xiong theorem (Theorem 2.1) covering both slow- and fast-decreasing sequences; the uncountable case, including the Erdős-Kunen-Mauldin theorem (Theorem 3.3), the Gallagher-Lai-Weber result that Cantor sets of positive Newhouse thickness are not measure universal (Theorem 3.6), and the extension to Cantor sets of positive Hausdorff dimension (Corollary 3.8); the topological variant, where Gallagher-Lai-Weber showed that no Cantor set is topologically universal and the Jung-Lai containment lemma gives another proof on the real line (Theorem 4.2); and a variant 'in the large' for increasing sequences (Theorems 5.2, 5.3, 5.6, 5.8), with metric number theory arguments in Section 6. Section 7 relates the original conjecture to these variants, including a Burgin-Goldberg-Keleti-MacMahon-Wang example (Theorem 7.1) of a positive-measure set in [0,1], with 0 as a Lebesgue density point, that contains no decreasing geometric sequence y b^{-n} (y != 0, b > 1). The abstract (p. 1) records that the conjecture remains open both for exponentially decaying sequences and for Cantor sets of zero Newhouse thickness and zero Hausdorff dimension, and p. 2 names the sequence 2^{-n} as the main open case; this is the conjecture of Problem 120.

Source: https://arxiv.org/abs/2412.11062.

Read status. Claims checked: the statements on the result pages below were read clause by clause on the printed pages. The survey's own proofs of Theorems 2.1, 3.6, 4.2 and 7.1 and of Corollary 3.8 were read but not checked step by step; Theorems 1.3 to 1.6 are quoted by the survey from other papers, which were not read.

Bears on. #120: Theorems 1.3, 1.4 and 1.5, part (1) of Theorem 2.1, Theorem 3.6 and Corollary 3.8 each answer the problem affirmatively for the infinite sets they cover (sublacunary decreasing sequences, sums of three infinite sets, sets with long slowly decaying chunks, Cantor sets of positive Newhouse thickness or positive Hausdorff dimension, and any set containing one of these); Theorem 1.6 is an almost-everywhere form for every infinite set; Theorem 4.2 is the topological analogue only; Theorem 7.1 avoids the geometric sequences y b^{-n} only without translation. None of them decides the sequence 2^{-n} or the general problem.

Results. Theorem 1.3 (p. 2, Eigen and Falconer); Theorem 1.4 (p. 2, Bourgain); Theorem 1.5 (p. 3, Kolountzakis); Theorem 1.6 (p. 3, Kolountzakis); Theorem 2.1 (p. 5, Feng, Lai and Xiong); Theorem 3.6 (p. 12, Gallagher, Lai and Weber); Corollary 3.8 (p. 13); Theorem 4.2 (p. 14, Jung and Lai); Theorem 7.1 (p. 22, Burgin, Goldberg, Keleti, MacMahon and Wang). The results of Sections 5 and 6 on the variant in the large concern increasing sequences in sets of large density and bear on no problem page of this corpus; they are summarized above and have no result pages.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.