Ternary expansions of powers of 2
Ternary expansions of powers of 2
复制标题
2 的三元展开式
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
J. Lagarias
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文献类型：
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作者：
J. Lagarias
Erdős 询问 2n 进行省略数字 2 的三元展开式的频率有多高。他推测这仅适用于有限多个 n 值。我们将这个问题概括为考虑两个离散动力系统的迭代。第一个考虑实数序列 xn (λ) = ⌊λ2n ⌋ 的截断三元展开，其中 λ > 0 是实数及其未截断版本，而第二个考虑序列 yn(λ) = λ2n 的 3-adic 展开，其中 λ 是 3-adic 整数。我们在这两种情况下都表明，具有无限多次省略数字 2 的迭代的初始值集在适当的意义上很小。对于每个非零初始值，我们在省略数字 2 的前 k 个迭代次数上获得渐近上限，即 k → ∞。我们还研究了有关 3-adic Cantor 集乘法平移交集的 Hausdorff 维数的辅助问题。
Erdős asked how frequently 2n has a ternary expansion that omits the digit 2. He conjectured that this holds only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first considers truncated ternary expansions of real sequences xn (λ) = ⌊λ2n ⌋, where λ > 0 is a real number, along with its untruncated version, whereas the second considers 3‐adic expansions of sequences yn(λ) = λ2n, where λ is a 3‐adic integer. We show in both cases that the set of initial values having infinitely many iterates that omit the digit 2 is small in a suitable sense. For each nonzero initial value we obtain an asymptotic upper bound as k → ∞ on the number of the first k iterates that omit the digit 2. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3‐adic Cantor sets.