Existence of words over a binary alphabet free from squares with mismatches

The paper is concerned with the problem of existence of periodic structures in words from formal languages. Squares (that is, fragments of the form $xx$, where $x$ is an arbitrary word) and $\Delta$-squares (that is, fragments of the form $xy$, where a word $x$ differs from a word $y$ by at most $\Delta$ letters) are considered as periodic structures. We show that in a binary alphabet there exist arbitrarily long words free from $\Delta$-squares with length at most $4\Delta+4$. In particular, a method of construction of such words for any $\Delta$ is given.