Simultaneous diophantine approximation with excluded primes

Given real numbers α<inf>1</inf>,...,α<inf><i>n</i></inf>, a simultaneous diophantine ε-approximation is a sequence of integers <i>P</i><inf>1</inf>,..., <i>P<inf>n</inf></i>, <i>Q</i> such that <i>Q</i> < 0 and for all <i>j</i> ∈ {1,...,<i>n</i>}, |<i>Q</i><inf>αj</inf>-<i>P</i><inf><i>j</i></inf>| ≤ ε. A simultaneous diophantine approximation is said to exclude the prime <i>p</i> if <i>Q</i> is not divisible by <i>p.</i> Given real numbers α<inf>1</inf>,...,α<inf><i>n</i></inf>, a prime <i>p</i> and ε > 0 we show that at least one of the following holds:<b>(a)</b>there is a simultaneous diophantine ε-approximation which excludes <i>p</i>, or<b>(b)</b>there exist <i>a</i><inf>1</inf>,...,<i>a<inf>n</inf></i> ∈ ℤ such that Σ<i>a</i><inf><i>j</i></inf>α<inf><i>j</i></inf> = 1/p + <i>t, t</i> ∈ ℤ and Σ|<i>a</i><inf><i>j</i></inf>|≤<i>n</i>^3/2|εNote that these two conditions are mutually nearly exclusive in the sense that in case (b) the <i>a<inf>j</inf></i> witness that there is no simultaneous diophantine ε/ (<i>n^3/2p</i>)-approximation excluding <i>p</i>. The proof method is Fourier analysis using results and techniques of Banaszczyk [Ban93].As an application we show that for <i>p</i> a prime and bounded <i>d/p</i> -- 1 the ring <i>ℤ/p^kℤ</i> contains a number all of whose <i>d</i>-th roots (mod <i>p^k</i>) 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.