Fekete’s lemma

For subadditive (or superadditive) sequences, the limit a_n/n exists and equals the infimum (or supremum); central for thermodynamic limits via (super/sub)additivity.

Fekete’s lemma

Statement (subadditive form)

Let $(a_n)_{n\ge 1}$ be a real sequence satisfying subadditivity

a_{m+n}\le a_m+a_n\qquad\text{for all }m,n\ge 1.

Then the limit

\lim_{n\to\infty}\frac{a_n}{n}

exists (possibly as $-\infty$ if one allows extended reals), and in the real-valued case it satisfies

\lim_{n\to\infty}\frac{a_n}{n}=\inf_{n\ge 1}\frac{a_n}{n}.

Statement (superadditive form)

If instead $(b_n)_{n\ge 1}$ is superadditive,

b_{m+n}\ge b_m+b_n\qquad\text{for all }m,n\ge 1,

then

\lim_{n\to\infty}\frac{b_n}{n}=\sup_{n\ge 1}\frac{b_n}{n}.

Key hypotheses

A sequence $(a_n)$ (or $(b_n)$ ) indexed by $n\ge 1$ .
Subadditivity $a_{m+n}\le a_m+a_n$ (or superadditivity $b_{m+n}\ge b_m+b_n$ ) for all $m,n$ .

Key conclusions

The “specific value” $a_n/n$ has a well-defined asymptotic limit under subadditivity, computable as an infimum.
The analogous “specific value” $b_n/n$ has a limit under superadditivity, computable as a supremum.

Proof idea / significance

Fix $n$ and write $m=qn+r$ with $0\le r<n$ . Subadditivity gives

a_m \le q\,a_n + a_r,

hence

\frac{a_m}{m}\le \frac{q\,a_n}{qn+r}+\frac{a_r}{qn+r}.

Letting $m\to\infty$ (so $q\to\infty$ ) yields $\limsup_{m\to\infty} a_m/m \le a_n/n$ . Since $n$ is arbitrary, $\limsup \le \inf_n a_n/n$ , while the reverse inequality $\inf_n a_n/n \le \liminf a_n/n$ is immediate from the definition of infimum, forcing the limit to exist and equal the infimum.

Thermodynamic significance: Fekete’s lemma is the standard mechanism behind existence of thermodynamic limits derived from subadditivity of log partition functions and superadditivity of entropy . Concretely, it underpins results like existence of the thermodynamic-limit pressure , where $a_n$ is typically (minus) $\log Z$ for a finite system and the limit defines the intensive pressure (or free-energy density).