The Falconer distance conjecture asserts that a compact set 
with Hausdorff dimension greater than 
has a distance set 
of positive Lebesgue
measure, where 
and 
is the Euclidean norm
(Iosevich and Yavicoli 2026).
Iosevich and Yavicoli (2026) proved that some absolute constant 
satisfies 
for every analytic
set 
with 
.
This is a Hausdorff dimension bound rather
than a proof of the conjecture.
OpenAI (2026) announced a proof in every integer dimension 
for compact sets 
with 
.
The claimed conclusion is positive Lebesgue measure
of the Euclidean distance set, rather than only a
lower Hausdorff dimension bound. The strict
threshold is retained and no endpoint or pinned-distance
conclusion is asserted here. The release proofs have not
been independently verified here.
Cheng et al. (2026) report a quantitative planar result. Let 
be compact, with
generalized diameter 
, supporting a probability
measure 
such that
 |
for every 
and 
,
where 
,
,
and 
is an open disk. They claim
 |
for every fixed 
, with 
depending only on 
and 
denoting Lebesgue
measure. In particular, their result would give positive Lebesgue
measure whenever 
. It does not assert the endpoint 
, a pinned-distance conclusion, or sharp constants.
The authors develop the OpenAI multiscale approach and report AI assistance with
local mathematical reasoning (Cheng et al. 2026). Their claimed proof
has not been independently verified here.
