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.
Cheng et al. (2026) report a quantitative planar result. Let 
be compact, with
generalized diameter 
, supporting a probability
measure 
such that
 |
(1)
|
for every 
and 
, where 
, 
,
and 
is an open disk. They claim
 |
(2)
|
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 (Cheng et al. 2026).
Burstein et al. (2026) report a strengthening of the positive-measure conclusion. For compact 
with 
and 
,
they construct a probability measure 
supported on 
and a number 
such that 
,
a Besov space, and
 |
(3)
|
where 
is a constant, 
,
and 
is the Fourier transform of 
. The probability
density function also belongs to 
for some 
.
These assertions concern a selected weighted distance measure, not every distance
measure or a fixed pinned distance set.
