Gruber P. Convex and Discrete Geometry

386 Geometry of Numbers
To show this, it is sufficient to show that there are unique integers u1,...,ud−1, with
0 ≤ u1,...,ud−1 < p, such that, for suitable integers k1,...,kd−1, k, wehave
v1 = ku1 + k1 p
.....................
vd−1 = kud−1 + kd−1 p
vd = k
or
v1 ≡ ku1 mod p
...................
vd−1 ≡ kud−1 mod p
vd = k.
Since vd = k is not an integer multiple of p, these congruences uniquely determine
integers u1,...,ud−1 with 0 ≤ u1,...,ud−1 < p, concluding the proof of (1). An
immediate consequence of (1) is as follows:
(2) The set Y = � v ∈ Z d : vd �= 0, ±p, ±2p,... � ⊆ Z d is the disjoint union
of the p d−1 sets L(p, u) ∩ Y : 0 ≤ u1,...,ud−1 < p.
Since J is Jordan measurable and thus bounded, and since V (J) 0 consider the hyperplanes
Hn ={x : xd = n/λ d−1 }, n = 0, ±1,...,parallel to E d−1 = H0. Letv(·) denote
(d − 1)-dimensional measure. Since J is Jordan measurable and compact, Fubini’s
theorem for Riemann integrals and the definition of 1-dimensional Riemann integrals
show that we may choose λ so large that the following statements hold: