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 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 probability
measure satisfying 
. Write 
and 
, where 
is Shannon entropy and
is the distribution of the composition
of 
independent maps
sampled from 
.
Kittle and Kogler (2026) reported the generalized dimension formula
 |
Here 
is exact-dimensional, meaning that ![nu([x-epsilon,x+epsilon])=epsilon^(dimnu+o(1))](/images/equations/ExactOverlapsConjecture/Inline15.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. Their
accompanying Lean formalization has not been independently checked here.
