Family 167CombinatoricsLeanDigested

Planar distinct distances and unit-distance bounds

Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but

o(n)

points of any n-point planar set determine at least

n^{1-\varepsilon}

distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by

O(n^{4/3-\delta})

for an absolute δ > 0.

Manuscripts in this family · 2

Sep 23, 2026·paper.pdfLean

We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but

o(n)

points of any n-point planar set determine at least

n^{1-\varepsilon}

distinct nonzero distances.

Full abstract

Sep 23, 2026·53 pp·paper.pdfDigested

We prove a power saving for the planar unit-distance problem: for some absolute

\beta\lt 4/3

, every set of n points in the Euclidean plane determines

O(n^\beta)

unordered pairs at unit distance.

Full abstract