Article
On Localization in Kronecker’s Diophantine Theorem
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Abstract
Using a probabilistic approach, we extend for general ℚ-linearly independent sequences a result of Tura´n concerning the sequence (log 𝑝ℓ), 𝑝ℓ being the ℓ-th prime. For instance let 𝜆1, 𝜆2, . . . be linearly independent over ℚ. We prove that there exists a constant 𝐶0 such that for any positive integers 𝑁 and 𝜔, if 𝑇 > 4𝜔 log 𝑁𝜔 𝑁 /Ξ, where ( 𝐶0 √ 𝐶0 ) then to any reals 𝑑, Ξ 𝛽1 = , . ∣. ∣𝑢 𝑢 . 1 𝑘 , 𝜆∣ 𝛽 ≤ 1 𝑁 𝑢 + 6 𝑘 𝜔 , .. m i . l n + o c t i g o e 𝑢 n ( g r 𝑁 𝑁 e r r e 𝜆 𝜔 s s 𝑁 / p 𝐶 ∣o = ∕ 0 n 0 ) ds 1≤ a ∑ 𝑘 r ≤ e 𝑁 al 𝜆 𝑡 𝑘𝑢 ∈ 𝑘[𝑑, 𝑑 + 𝑇 ] such that sup𝑁 𝑗=1 𝑡𝜆𝑗 − 𝛽𝑗 ≤ 1/𝜔.
Keywords
Diophantine approximation, discrete random variable, characteristic function.
Citation
Weber, M. (2009). On localization in kronecker’s diophantine theorem. Uniform Distribution Theory, 4(1), 97–116.
M. Weber, “On localization in kronecker’s diophantine theorem,” Uniform Distribution Theory, vol. 4, no. 1, pp. 97–116, 2009.
Weber M. On localization in kronecker’s diophantine theorem. Uniform Distribution Theory. 2009;4(1):97–116.
Weber, M. (2009), ‘On localization in kronecker’s diophantine theorem’, Uniform Distribution Theory, 4(1), pp. 97–116.
Weber, Michel. “On Localization in Kronecker’s Diophantine Theorem.” Uniform Distribution Theory, vol. 4, no. 1, 2009, pp. 97–116.
Weber, Michel. “On Localization in Kronecker’s Diophantine Theorem.” Uniform Distribution Theory 4, no. 1 (2009): 97–116.
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Published by: Engineering Journals
