Abstract
We refine a result of Languasco and Zaccagnini (J Number Theory 132:3016–3028, 2012) on Diophantine approximation with two primes and powers of two. We improve the upper bound on the least value \(s_0\) which ensures that for all \(s\ge s_0\), there exists an approximation to any real number by values of the form \(\lambda _1p_1+\lambda _2p_2+\mu _1 2^{m_1}+\dots +\mu _s 2^{m_s}\), where \(p_1\), \(p_2\) are primes; \(m_1\), \(\dots \), \(m_s\) are positive integers; \(\lambda _1/\lambda _2\) belongs to a certain subset of negative irrational numbers and \(\lambda _1/\mu _1\), \(\lambda _2/\mu _2\in \mathbb {Q}\).
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The author is supported by the China Scholarship Council.
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Wang, Y. Diophantine approximation with two primes and powers of two. Ramanujan J 39, 235–245 (2016). https://doi.org/10.1007/s11139-015-9690-z
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DOI: https://doi.org/10.1007/s11139-015-9690-z