  Ufnarovski remarked in 1990 that it is unknown whether any finitely presented
associative algebra of linear growth is automaton, that is, whether the set of
normal words in the algebra form a regular language. If the algebra is graded,
then the rationality of the Hilbert series of the algebra follows from the
affirmative answer to Ufnarovski's question. Assuming that the ground field has
a positive characteristic, we show that the answer to Ufnarovskii's question is
positive if and only if the basic field is an algebraic extension of its prime
subfield. Moreover, in the "only if" part we show that there exists a finitely
presented graded algebra of linear growth with irrational Hilbert series. In
addition, over an arbitrary infinite basic field, the set of Hilbert series of
the quadratic algebras of linear growth with $5$ generators is infinite.
  Our approach is based on a connection with the dynamical Mordell--Lang
conjecture. This conjecture describes the intersection of an orbit of an
algebraic variety endomorphism with a subvariety. We show that the positive
answer to Ufnarovski's question implies some known cases of the dynamical
Mordell--Lang conjecture. In particular, the positive answer for a class of
algebras is equivalent to the Skolem--Mahler--Lech theorem which says that the
set of the zero elements of any linear recurrent sequence over a zero
characteristic field is the finite union of several arithmetic progressions. In
particular, the counter-examples to this theorem in the finite characteristic
case give examples of algebras with irrational Hilbert series.
