Study On The Length Of Consecutive Zeros In The β-expansion Of Real Numbers

For any real numbers x ∈[0,1]and β>1,let rn(x,β)be the maximal length of consecutive 0s in the first n digits of the beta-expansion of x in base β.The run-length function rn(x,β)has been well studied for a fixed base β or a fixed real number x=1.In this thesis,we will use the Borel-Cantelli lemma and the mass distribution principle to explore the asymptotic behavior of the run-length function rn(x,β)with n→∞ for a general x ∈(0,1).we first prove that for any real number x ∈(0,1),the set(?) is of full Lebesgue measure in(1,+∞).After considering the measure property of the set,we prove that for any real numbers x ∈(0,1)and 0 ≤a ≤b ≤+∞,the Hausdorff dimension of the set (?)is 1.Further,we obtain when the run-length function rn(x,β)tends to infinity at a more general speed φ(n),for any real numbers x ∈(0,1)and 0≤a≤b≤+∞,the Hausdorff dimension of the set (?)is also 1,where φ:N→R+ is an arbitrarily strictly increasing function with φ(n)tending to +∞ and φ(n)/n decreasing to 0 as n→+∞.Furthermore,we also determine the Hausdorff dimension of the set (?)for any real numbers x ∈(0,1)and 0 ≤c≤d≤1.