The exact overlaps conjecture asserts that a drop in the Hausdorff dimension of a self-similar subset
of the real line below its expected value can occur
only through exact coincidences of compositions in
its defining iterated function system.
Let 
be a finite family of distinct contracting similarities
with 
. An exact overlap occurs when two distinct
words of the same length in the 
define the same map.
For positive probabilities 
summing to 1, let 
and let 
be the unique self-similar
measure satisfying 
,
where 
denotes the pushforward measure under 
. Write 
and 
, where 
is Shannon entropy computed
using natural logarithms and 
is the distribution
of the composition of 
independent maps sampled from 
.
Kittle and Kogler (2026) reported the generalized dimension formula
 |
Here 
is an exact-dimensional measure, meaning
that ![nu([x-epsilon,x+epsilon])=epsilon^(dimnu+o(1))](/images/equations/ExactOverlapsConjecture/Inline17.svg)
for 
-almost
every 
as 
.
In particular, if there are no exact overlaps, the self-similar attractor 
has
 |
where 
. Kittle and Kogler (2026) credit GPT-6
Astra with the initial proof and the key entropy inequality.
They report rewriting the argument and supplying a different final deduction, and
they take responsibility for the mathematics. They supply an accompanying Lean formalization.
