On the Powers of Some Transcendental Numbers
On the Powers of Some Transcendental Numbers
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论某些超越数的幂
DOI：
10.1017/s0004972700039782
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发表时间：
2007
影响因子：
0.7
通讯作者：
A. Dubickas
中科院分区：
文献类型：
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作者：
A. Dubickas
我们构造一个超越数α，它的幂αn！，n = 1，2，3，...，模1在区间[0，1]中处处稠密。类似地，对于任何正数序列δ =（δn）∞n=1，我们找到一个超越数α = α（δ），使得不等式{αn} < δn对无穷多个n ∈ N成立，不管序列δ收敛到零的速度有多快。最后，对任意真实的数列（rn）∞n=1和任意正数数列（δn）∞n=1，构造了一个正整数递增数列（qn）∞n=1和一个数α > 1，使得对任意n ≥ 1，< δn.
We construct a transcendental number α whose powers αn!, n = 1, 2, 3,…, modulo 1 are everywhere dense in the interval [0, 1]. Similarly, for any sequence of positive numbers δ = (δn)∞n=1, we find a transcendental number α = α(δ) such that the inequality {αn} < δn holds for infinitely many n ∈ N, no matter how fast the sequence δ converges to zero. Finally, for any sequence of real numbers (rn)∞n=1 and any sequence of positive numbers (δn)∞n=1, we construct an increasing sequence of positive integers (qn)∞n=1 and a number α > 1 such that ‖αqn – τn‖ < δn for each n ≥ 1.