The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
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Given real numbers α1,...,αn, a simultaneous diophantine ε-approximation is a sequence of integers P1,..., Pn, Q such that Q 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.