摘要
For any x ∈ (0, 1] (except at most countably many points), there exists a unique sequence {dn(x)}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1(x)(d1(x)-1)……dj-1(x)(dj-1(x)-1)dj(x). The dexter infinite series expansion is called the Liiroth expansion of x. This paper is con- cerned with the size of the set of points x whose digit sequence in its Liiroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary com- mon difference. More precisely, we determine the Hausdorff dimension of the above set.
For any x ∈ (0, 1] (except at most countably many points), there exists a unique sequence {dn(x)}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1(x)(d1(x)-1)……dj-1(x)(dj-1(x)-1)dj(x). The dexter infinite series expansion is called the Liiroth expansion of x. This paper is con- cerned with the size of the set of points x whose digit sequence in its Liiroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary com- mon difference. More precisely, we determine the Hausdorff dimension of the above set.
基金
supported by NSFC(11326206,11426111)