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

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

Then the limit

lim⁡n→∞ann \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→∞ann=inf⁡n≥1ann. \lim_{n\to\infty}\frac{a_n}{n}=\inf_{n\ge 1}\frac{a_n}{n}.

Statement (superadditive form)

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

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

then

lim⁡n→∞bnn=sup⁡n≥1bnn. \lim_{n\to\infty}\frac{b_n}{n}=\sup_{n\ge 1}\frac{b_n}{n}.

Key hypotheses

  • A sequence (an)(a_n) (or (bn)(b_n)) indexed by n≥1n\ge 1.
  • Subadditivity am+n≤am+ana_{m+n}\le a_m+a_n (or superadditivity bm+n≥bm+bnb_{m+n}\ge b_m+b_n) for all m,nm,n.

Key conclusions

  • The “specific value” an/na_n/n has a well-defined asymptotic limit under subadditivity, computable as an infimum.
  • The analogous “specific value” bn/nb_n/n has a limit under superadditivity, computable as a supremum.

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