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An important problem in combinatorial noncommutative algebra is tocharacterize the growth functions of finitely generated algebras (equivalently,semigroups, or hereditary languages).  The growth function of every finitely generated, infinite-dimensional algebrais increasing and submultiplicative. The question of to what extent thesenatural necessary conditions are also sufficient -- and in particular, whetherthey are sufficient at least for sufficiently rapid functions -- was posed andstudied by various authors and has attracted a flurry of research.  While every increasing and submultiplicative function is realizable as agrowth function up to a linear error term, we show that there exist arbitrarilyrapid increasing submultiplicative functions which are not equivalent to thegrowth of any algebra, thus resolving the aforementioned problem and settling aquestion posed by Zelmanov (and repeated by Alahmadi-Alsulami-Jain-Zelmanov).These can be interpreted as `holes' in the space of growth functions,accumulating to exponential functions in the order topology.  We show that there exist monomial algebras and hereditary languages whosegrowth functions encode the existence of non-prolongable words, and algebraswhose growth functions encode the existence of nilpotent ideals (in the gradedcase). This negatively solves another conjecture ofAlahmadi-Alsulami-Jain-Zelmanov in the graded case.

Gaps and approximations in the space of growth functions