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On the Powers of Some Transcendental Numbers

  • Artūras Dubickas (a1)

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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.

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References

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On the Powers of Some Transcendental Numbers

  • Artūras Dubickas (a1)

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