Computation of highly regular nearby points

We call a vector x∈Rⁿ highly regular if it satisfies <x,m>=0 for some short, non-zero integer vector m where <...> is the inner product. We present an algorithm which given x∈Rⁿ and α∈N finds a highly regular nearby point x´ and a short integer relation m for x´. The nearby point x´ is `good´ in the sense that no short relation m¯ of length less than α/2 exists for points x¯ within half the x´-distance from x. The integer relation m for x´ is for random x up to an average factor 2^α/2 a shortest integer relation for x´. Our algorithm uses, for arbitrary real input x, at most O(n⁴(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n ⁵+n³(log α)²+log(||qx||²)) many bits where q is the common denominator for x