The factor complexity 
of a finite or infinite word 
, also called its subword complexity,
is the number of distinct contiguous factors of length
occurring in 
.
The Morse-Hedlund theorem states that a right-infinite word is ultimately
periodic iff 
for some positive
integer 
. Consequently, a right-infinite word that is not ultimately
periodic has factor complexity at least 
for every positive integer 
. Sturmian sequences
are precisely the aperiodic right-infinite binary words attaining 
for every positive
integer 
.
Among right-infinite ternary words with factor complexity 
for every positive integer 
, the minimum possible critical
exponent is 
(Currie 2026).
For the base-
digit expansion of a real algebraic
number 
that is irrational, write the factor complexity
as 
. Stephan (2026) proved that for every integer 
, every 
, and every 
,
 |
for infinitely many 
, improving the range 
of Bugeaud and Evertse (2008). This is not an eventual
lower bound and does not establish that 
is a normal number. Stephan
(2026) obtained the analogous bound for the word of partial
quotients in the continued fraction of a real
algebraic number of degree at least three. The
paper reports Lean formalizations.
and Minimal Critical Exponent." Elec. J. Combin. 33,
P3.31, 2026. 
in Three Complexity Bounds: Integer Bases, Continued Fractions,
and the 
Steering Word." 22 Oct 2026.