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On Simultaneously Badly Approximable Numbers
Journal of the London Mathematical Society, 2002For any \(i,j>0\) with \(i+j=1\) define the set \(B(i,j)\) to be the set of pairs of reals \((\alpha,\beta)\) with \[ \max\{\|q\alpha\|^{1/i}\|q\beta \|^{1/j}\}>C(\alpha,\beta)/q \] for all \(q\geq 1\) and some positive constant \(C(\alpha,\beta)\). For \(i=0\) set \(B(0,1)\) to be the set of pairs \((\alpha,\beta)\) where \(\alpha\) is any real and \(\
Pollington, Andrew, Velani, Sanju
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On Approximating the Achromatic Number
SIAM Journal on Discrete Mathematics, 2001Summary: The achromatic number problem is to legally color the vertices of an input graph with the maximum number of colors, denoted \(\psi^*\), so that every two color classes share at least one edge. This problem is known to be NP-hard. For general graphs we give an algorithm that approximates the achromatic number within a ratio of \(O(n\cdot \log ...
Kortsarz, Guy, Krauthgamer, Robert
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RATIONAL APPROXIMATIONS TO ALGEBRAIC NUMBERS
Mathematika, 1955This important paper contains a proof of the long conjectured theorem: ``If \(\alpha\) is an algebraic irrational number, and if there are infinitely many fractions \(h/q\) with \(\vert \alpha - h/q\vert \le q^{-\kappa}\), \(q > 0\), \((h, q) = 1\), then \(\kappa\le 2\).'' Much of the proof runs on classical lines.
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Recursive Approximability of Real Numbers
Mathematical Logic Quarterly, 2002zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Entropy Numbers and Approximation Numbers in Function Spaces, II
Proceedings of the London Mathematical Society, 1989This paper continues the study of entropy and approximation numbers related to compact embeddings between scales of Besov type function spaces \(B^ s_{p,q}\). In a previous paper [Proc. London Math Soc., III. Ser. 58, No. 1, 137-152 (1989; Zbl 0629.46034)], the authors obtained estimates from above for the entropy numbers \(e_ k\) and approximation ...
Edmunds, D. E, Triebel, H.
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2003
As we have already mentioned, the numbers we know the best are the integers. These occur very infrequently among the real numbers. Quotients of these, the rational numbers, occur much more often, but as we will see, in a certain sense, they make up only an insignificant part of the real numbers.
Paul Erdős, János Surányi
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As we have already mentioned, the numbers we know the best are the integers. These occur very infrequently among the real numbers. Quotients of these, the rational numbers, occur much more often, but as we will see, in a certain sense, they make up only an insignificant part of the real numbers.
Paul Erdős, János Surányi
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Diophantine Approximation by Prime Numbers
Journal of the London Mathematical Society, 1991The author proves: Suppose that \(\lambda_ 1,\lambda_ 2,\lambda_ 3\) are non-zero real numbers not all of the same sign, that \(\alpha\) is real and that \(\lambda_ 1/\lambda_ 2\) is irrational. Then, for any \(\varepsilon>0\), there are infinitely many ordered triples of primes \(p_ 1,p_ 2,p_ 3\) for which \[ |\alpha+\lambda_ 1p_ 1+\lambda_ 2p_ 2 ...
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