------ Dan writes: << so there is no "best" value of X. I'll leave finding "good" values of X to the number theorists, once "good" is defined.
------ Victor writes: << That's true, but one can ask for what the best value of X is given that all elements of X are bounded in absolute value by H. There is an algorithm due to Hastad, Just, Lagarias and Schnorr that is designed to answer the posed question. When looked at the right way it's very close to the L^3 lattice reduction algorithm.
--------------------- Many thanks Victor for this reference: its title looks exactly what I am searching. You are right in your answer to Dan. "Good" means using smallest integers abs(x_i) < H allowing an asked precision. For example: k_1 = pi k_2 = e Using the continued fraction algorithm on pi/e, we will find good x_1 and x_2: k_1*x_1 + k_2*x_2 ~ 0 ------------------------------ e*7 + pi*(-6) = 0.1784 e*15 + pi*(-13) = -0.0664 e*37 + pi*(-32) = 0.0454 e*52 + pi*(-45) = -0.0210 e*141 + pi*(-122) = 0.0034 ... Now, what would be solutions with: k_1 = pi k_2 = e k_3 = sqrt(2) k_1*x_1 + k_2*x_2 + k_3*x_3 ~ 0 --------------------------------- ??? ??? ??? Probably the method explained in the paper will be able to produce some "good" values. Christian.