On gapped repeats in a cyclic Fibonacci word

The authors study how certain sequences of letters called words match up when you look at parts of them starting from different positions in a looped way (cyclic indices). They focus on the special Fibonacci word, which is built using a pattern related to the Fibonacci numbers. They figure out exactly when two starting points produce the same sequence of letters of a certain length. They also count how many such matching pairs exist within this cyclic Fibonacci word.