Problem 232

For $A\subset \mathbb{R}^2$ we define the upper density as

\overline{\delta}(A)=\limsup_{R\to \infty}\frac{\lambda(A \cap B_R)}{\lambda(B_R)},

where $\lambda$ is the Lebesgue measure and $B_R$ is the ball of radius $R$ .

Estimate

m_1=\sup \overline{\delta}(A),

where $A$ ranges over all measurable subsets of $\mathbb{R}^2$ without two points distance $1$ apart. In particular, is $m_1\leq 1/4$ ?

Source: erdosproblems.com/232

Geometry
Distances

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Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. The site's label answers the particular question: Ambrus, Csiszárik, Matolcsi, Varga and Zsámboki proved $m_1\le0.247<1/4$ , which answers the site's question and proves Erdős's conjecture $m_1<1/4$ [Er85], read with the circle's area as the normalization. The conjecture as printed (p. 4) divides the measure by $R^2$ rather than by the area $\pi R^2$ , and in that form it is false. The estimate of $m_1$ itself remains between Croft's lower bound $0.22936$ and this upper bound. The result is recorded in claims/ as an accepted claim.

ACMVZ23: G. Ambrus, A. Csiszárik, M. Matolcsi, D. Varga and P. Zsámboki, The density of planar sets avoiding unit distances, Math. Program. 207 (2024), 303–327, doi:10.1007/s10107-023-02012-9; arXiv:2207.14179.
Cr67: H. T. Croft, Incidence incidents, Eureka 30 (1967), 22–26.
Er85: P. Erdős, Problems and results in combinatorial geometry, in Discrete geometry and convexity (New York, 1982), Ann. New York Acad. Sci. 440 (1985), 1–11.
Mo66: L. Moser, Poorly formulated unsolved problems of combinatorial geometry. Mimeographed (1966).

Problem 232

Status

Proof claims

Discussion