Consecutive Power Occurrences in Sturmian Words

We show that every Sturmian word has the property that the distance between consecutive ending positions of cubes occurring in the word is always bounded by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>10</mml:mn> </mml:math> and this bound is optimal, extending a result of Rampersad, who proved that the bound <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>9</mml:mn> </mml:math> holds for the Fibonacci word. We then give a general result showing that for every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> there is a natural number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> , depending only on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> , such that every Sturmian word has the property that the distance between consecutive ending positions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>e</mml:mi> </mml:math> -powers occurring in the word is uniformly bounded by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> .