Filho 2019 counterexample conjecture larman rogers sets avoiding

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construction_p1: De Oliveira Filho and Vallentin's set S_n, two opposite lens-shaped pieces of the unit ball of R^n with no two points at distance 1, whose volume exceeds (1/2)^n times the volume of the ball for every n at least 2, so refuting Conjecture 1 of Larman and Rogers (1972).

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Fernando Mário de Oliveira Filho, Frank Vallentin, A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1, Mathematika 65 (2019), 785--787, doi:10.1112/S0025579319000160; arXiv:1808.07299. The copy read for this card is arXiv:1808.07299v2 (11 March 2019, the final version); page numbers refer to it. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1808.07299), every other right reserved.

The note refutes Conjecture 1 of Larman and Rogers, which asserted that a closed subset of a unit ball avoiding distance 1 has Lebesgue measure less than (1/2)^n times that of the ball, a bound attained in the limit by an open ball of radius 1/2. The counterexample is explicit and elementary: with a = (1 + sqrt 10)/6, let T_n be the set of x in R^n with x_1 > 1/2, |x - a e_1| < 1/2 and |x| < 1, and set S_n = T_n union -T_n; S_n avoids distance 1, and its two symmetric caps beat the single half-radius ball. Direct integration gives vol S_2 / vol B_2 = 0.2848... and vol S_3 / vol B_3 = 0.1563..., and the concentration of the ball's volume near the equator yields the asymptotic vol S_n / vol B_n = (2 - o(1)) (1/2)^n, strictly above (1/2)^n for all n >= 15, the remaining dimensions being checked directly. The paper explains that one of Larman and Rogers's motivations was the related L. Moser conjecture popularized by Erdos, that the upper density of a distance-one-avoiding measurable subset of the plane is below 1/4 (open when the paper was written, since proved by Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki, who showed m_1 <= 0.247); their conjecture would have given only the weaker bound at most 1/4, and so would not have implied Moser's. For problem 1070 the paper was archived in a citation sweep to disambiguate the statement-cited Larman-Rogers reference: it refutes their separate fixed-unit-ball volume conjecture in every dimension n >= 2 and gives no new global bound on m_1, so it is not itself a resolution of problem 1070.

Source: https://arxiv.org/abs/1808.07299.

Bears on. #1070: indirect. The paper recalls Moser's conjecture, which it calls still open, that a measurable planar set avoiding distance 1 has upper density less than 1/4 (the supremum of these densities is the m_1 of the problem page's bound f(n) >= m_1 n), and refutes only Larman and Rogers's local conjecture for a unit ball, which in the plane would have given m_1 <= 1/4 but even if true would not have implied Moser's; it gives no bound on m_1 or f(n).

Results. The construction (p. 1, unnumbered; relation (1) and the range n >= 15 on p. 2): for every n >= 2 the set S_n = T_n union -T_n, with T_n cut from the open unit ball by the open ball of radius 1/2 about a e_1 and the halfspace x_1 > 1/2, where a = (1 + sqrt 10)/6, avoids distance 1 and has volume greater than (1/2)^n vol B_n; vol S_2 / vol B_2 = 0.2848... and vol S_3 / vol B_3 = 0.1563...; and vol S_n / vol B_n = (2 - o(1))(1/2)^n. Closed inner approximations of S_n meet the conjecture's closedness hypothesis (footnote 1, p. 1).

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