  Let $A_q$ be a $q$-letter alphabet and $w$ be a right infinite word on this
alphabet. A subword of $w$ is a block of consecutive letters of $w$. The
subword complexity function of $w$ assigns to each positive integer $n$ the
number $f_w(n)$ of distinct subwords of length $n$ of $w$.
  The gap function of an infinite word over the binary alphabet $\{0,1 \}$
gives the distances between consecutive 1's in this word. In this paper we
study infinite binary words whose gap function is injective or "almost
injective". A method for computing the subword complexity of such words is
given. A necessary and sufficient condition for a function to be the subword
complexity function of a binary word whose gap function is strictly increasing
is obtained.
