THE MIXED SCHMIDT CONJECTURE IN THE THEORY OF DIOPHANTINE APPROXIMATION
THE MIXED SCHMIDT CONJECTURE IN THE THEORY OF DIOPHANTINE APPROXIMATION
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丢番图近似理论中的混合施密特猜想
DOI：
10.1112/s0025579311002075
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发表时间：
2011
期刊：
影响因子：
0.8
通讯作者：
Badziahin D
中科院分区：
文献类型：
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作者：
Badziahin D
设=(dn)∞n=1是一个dn≥2的整数序列，设（i,j）是一对i+j=1的严格正数。我们证明了x∈x的集合存在某常数c(x)≧0使得1 / 4获胜（在施密特博弈的意义上）。因此，任何可数的这样的集合的交点都是全维的。这反过来又在de mathen - teuli<e:1>猜想（也称为“混合Littlewood猜想”）的框架内建立了Schmidt猜想的自然类比。
Let 𝒟=(dn)∞n=1 be a sequence of integers with dn≥2, and let (i,j) be a pair of strictly positive numbers with i+j=1. We prove that the set of x∈ℝ for which there exists some constant c(x)≧0 such that is one-quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. This, in turn, establishes the natural analogue of Schmidt’s conjecture within the framework of the de Mathan–Teulié conjecture, also known as the “mixed Littlewood conjecture”.