3.3.4 THE SPECTRAL TEST 93 where l4) = (Web hosting faq)
3.3.4 THE SPECTRAL TEST 93 where l4) = k(o, c, (1 + a)c, . . . , (1 + a + . + cP )c) (6) is a constant vector. The variable k1 is redundant in this representation of L, be- cause we can change (5, ki, Its, . . . , k,) to (z+kim, 0, ks –&I,. . . , kt –at- kl), reducing kl to zero without loss of generality. Therefore we obtain the compara- tively simple formula L={Vo+ylK+y2V2+~~~+ytVt I integeryl,ya,…,yt}, (7) where vl = ~(l,u,u2 ,.. , CL- ); (8) Vs=(O,l,O )…, O), vs=(o,o,1,. .) O), . ..) Vt=(O,O,O ) .) 1). (9) The points (zi, x2,. . . , xt) of L that satisfy 0 5 x3 < 1 for all j are precisely the m points of our original set (2). Note that the increment c appears only in Vo, and the effect of V. is merely to shift all elements of L without changing their relative distances; hence c does not affect the spectral test in any way, and we might as well assume that Vi = 640,. . . , 0) when we are calculating LQ. When Vo is the zero vector we have a so-called lattice of points Lo={~IK+Y~V~+.~.+Y~K I integeryl,yz,...,yt), (10) and our goal is to study the distances between adjacent (t -1)-dimensional hyperplanes, in families of parallel hyperplanes that cover all the points of Lo. A family of parallel (t -1)-dimensional hyperplanes can be defined by a nonzero vector U = (ui, . . . , Ut) that is perpendicular to all of them; and the set of points on a particular hyperplane is then {(Xl,. . . 7 xt) I 21% + . . * + xtw = 4 >, (11) where 4 is a different constant for each hyperplane in the family. In other words, each hyperplane is the set of all X for which the dot product X . U has a given value 4. In our case the hyperplanes are all separated by a fixed distance, and one of them contains (O,O, . . . ,O); hence we can adjust the magnitude of U so that the set of all integer values c~ gives all the hyperplanes in the family. Then the distance between neighboring hyperplanes is the minimum distance from (O,O,. . . , 0) to the hyperplane for o = 1, namely Cauchy s inequality (cf. exercise 1.2.3-30) tells us that (13)
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