On self-matching within integer part sequences

With α irrational the graph of [jα] against integer j, displays interesting patterns of self-matching. This is best seen by comparing the Bernoulli (characteristic Sturmian) or difference sequence 〈βj〉, term by term with the Bernoulli sequence displaced by k terms 〈βj−k〉, where βj=[(j+1)α]−[jα].It is shown that the fraction of such self-matching is the surprisingly simple M(k)=max(|1−2{α}|,|1−2{kα}|).Of particular interest is the graph of M(k) against k as it is seen to exhibit an unexpected Moiré pattern obtained simply by folding the lower half of the graph of {kα} over the upper half.