Given real numbers α1,...,αn, a simultaneous diophantine ε-approximation is a sequence of integers P1,..., Pn, Q such that Q 0 and for all j ∈ {1,...,n}, |Qαj-Pj| ≤ ε. A simultaneous diophantine approximation is said to exclude the prime p if Q is not divisible by p. Given real numbers α1,...,αn, a prime p and ε 0 we show that at least one of the following holds:(a)there is a simultaneous diophantine ε-approximation which excludes p, or(b)there exist a1,...,an ∈ ℤ such that Σajαj = 1/p + t, t ∈ ℤ and Σ|aj|≤n3/2|εNote that these two conditions are mutually nearly exclusive in the sense that in case (b) the aj witness that there is no simultaneous diophantine ε/ (n3/2p)-approximation excluding p. The proof method is Fourier analysis using results and techniques of Banaszczyk [Ban93].As an application we show that for p a prime and bounded d/p -- 1 the ring ℤ/pkℤ contains a number all of whose d-th roots (mod pk) are small.We generalize the result to simultaneous diophantine ε-approximations excluding several primes and consider the algorithmic problem of finding, in polynomial time, a simultaneous diophantine ε-approximation excluding a set of primes.
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机译:给定实数α 1 INF>,...,α n I> INF>,同时双色子烷ε逼近是整数 P < / I> 1 INF>,..., P n INF> I>, Q I>,这样 Q I > <0且对于所有 j I>∈{1,..., n I>},| Q I> αj INF>- P I> j I> INF> | ≤ε。如果 Q I>不能被 p整除,则称同时双色子素近似可排除素数 p I>。给定实数α 1 INF>,...,α n I> INF>,素数 p I>和ε> 0,我们证明至少有以下一项成立: B>(a) B>同时存在双色子素ε逼近,其中不包括 p I>或(b) B>存在 a I> < INF> 1 INF>,..., a n INF> I>∈ℤ使得Σ a I> j I> INF>α j I> INF> = 1 / p + t,t I>∈ℤ和Σ| a I> j I> INF> |≤ n I> 3/2 SUP> |ε注意,在情况(b) a j INF> I>见证没有同时的双色子素ε/( n 3/2 SUP> p I> )-近似值,不包括 p I>。证明方法是使用Banaszczyk [Ban93]的结果和技术进行的傅里叶分析。作为一个应用,我们证明了对于 p I>质数和有界 d / p I>-1环ℤ/ p k SUP>ℤ I>包含许多其所有 d I>根的数(mod p k SUP> I>)很小。我们将结果推广为排除几个素数的同时双色子ε-逼近,并考虑了在多项式时间内找到不包含素数的同时二色子ε-逼近的算法问题。
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