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...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.展开更多
For Oppenheim series epansions, the authors of [7] discussed the exceptional sets Bm={x∈（0,1]：1〈dj（x）/h（j-1）（d（j-1）（x））≤m for any j ≥2} In this paper, we investigate the Hausdorff dimension of a kind o...For Oppenheim series epansions, the authors of [7] discussed the exceptional sets Bm={x∈（0,1]：1〈dj（x）/h（j-1）（d（j-1）（x））≤m for any j ≥2} In this paper, we investigate the Hausdorff dimension of a kind of exceptional sets occurring in alternating Oppenheim series expansion. As an application, we get the exact Hausdorff dimension of the-set in Luroth series expansion, also we give an estimate of such dimensional number.展开更多
It is well known that for almost all real number x, the geometric mean of the first n digits di（x） in the Lüroth expansion of x converges to a number K0 as n→∞. On the other hand, for almost all x, the arithm...It is well known that for almost all real number x, the geometric mean of the first n digits di（x） in the Lüroth expansion of x converges to a number K0 as n→∞. On the other hand, for almost all x, the arithmetric mean of the first n Lüroth expansion digits di（x） approaches infinity as n→∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin＇s inequalities, relating the 1/k-th powers of the k-th elementary symmetric means of n numbers for 1≤k≤n. In this paper, we investigate what happens to the means of Lüroth expansion digits in the limit as one moves f（n） steps away from either extreme. We prove sufficient conditions on f（n） to ensure divergence when one moves away from the arithmetic mean and convergence when one moves f（n） steps away from geometric mean.展开更多
文摘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.
文摘For Oppenheim series epansions, the authors of [7] discussed the exceptional sets Bm={x∈（0,1]：1〈dj（x）/h（j-1）（d（j-1）（x））≤m for any j ≥2} In this paper, we investigate the Hausdorff dimension of a kind of exceptional sets occurring in alternating Oppenheim series expansion. As an application, we get the exact Hausdorff dimension of the-set in Luroth series expansion, also we give an estimate of such dimensional number.
基金Supported by the National Natural Science Foundation of China(11271148)
文摘It is well known that for almost all real number x, the geometric mean of the first n digits di（x） in the Lüroth expansion of x converges to a number K0 as n→∞. On the other hand, for almost all x, the arithmetric mean of the first n Lüroth expansion digits di（x） approaches infinity as n→∞. There is a sequence of refinements of the AM-GM inequality, Maclaurin＇s inequalities, relating the 1/k-th powers of the k-th elementary symmetric means of n numbers for 1≤k≤n. In this paper, we investigate what happens to the means of Lüroth expansion digits in the limit as one moves f（n） steps away from either extreme. We prove sufficient conditions on f（n） to ensure divergence when one moves away from the arithmetic mean and convergence when one moves f（n） steps away from geometric mean.
