On cogrowth function of algebras and its logarithmical gap

Let \(A \cong k\langle X \rangle / I\) be an associative algebra. A finite word over alphabet \(X\) is \(I\){\it-reducible} if its image in \(A\) is a \(k\)-linear combination of length-lexicographically lesser words. An {\it obstruction} in a subword-minimal \(I\)-reducible word. A {\em cogrowth} function is number of obstructions of length \(\le n\). We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.