On self-matching within integer part sequences

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