Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  call a vector x/spl isin/R/sup n/ highly regular if it satisfies  =0 for some short, non-zero integer vector m where   is the inner product.  present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//spl isin/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 /spl alpha//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/sup /spl alpha//2/ a shortest integer relation for x'. Our algorithm uses, for arbitrary real input x, at most O(n/sup 4/(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n/sup 5/+n/sup 3/(log /spl alpha/)/sup 2/+log(/spl par/qx/spl par//sup 2/)) many bits where q is the common denominator for x. >
['Carsten Rössner', 'Claus-Peter Schnorr']
I am writing a research paper on a idea Computation of highly regular nearby points
[{"from": "human", "value": "I am writing a research paper on a idea Computation of highly regular nearby points"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  call a vector x/spl isin/R/sup n/ highly regular if it satisfies  =0 for some short, non-zero integer vector m where   is the inner product.  present an algorithm which given x/spl isin/R/sup n/ and /spl alpha//spl isin/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 /spl alpha//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/sup /spl alpha//2/ a shortest integer relation for x'. Our algorithm uses, for arbitrary real input x, at most O(n/sup 4/(n+log A)) many arithmetical operations on real numbers. If a is rational the algorithm operates on integers having at most O(n/sup 5/+n/sup 3/(log /spl alpha/)/sup 2/+log(/spl par/qx/spl par//sup 2/)) many bits where q is the common denominator for x. >"}]