Computing the measure of disks and angular sectors with respect to a standard den-sity of valuationr leads to the estimates of the main output distributions. We first present the main constants that will intervene in our results.
Constants of the Analysis
The measure of a disk of radiuscentered on the real axis equals 2A2.r/ rC2. The measure of a disk of radius tangent to the real axis equalsA1.r/ .2/rC2. Such measures involve constantsA1.r/; A2.r/, which are expressed with the ˇ law, already defined in (3.21) as
A1.r/WD p A.r/
.rC3=2/
.rC3/ ; A2.r/WD p 2A.r/
..rC1/=2/
.r=2C2/ : (3.49) For a triangle with basisa on the real axis and heighth, this measure equals A3.r/ a hrC1, and involves the constant
A3.r/WD 1 A.r/
1
.rC2/.rC1/: (3.50)
For.˛; ˇ/that belongs to the triangleT WD f.˛; ˇ/I0 < ˛; ˇ < 1; ˛Cˇ > 1g, we consider the continuous analogs of the configurations previously described:
Disks. We consider the figure obtained with three disksD˛; Dˇ; D˛Cˇwhen these disks satisfy the following: For anyı; 2 f˛; ˇ; ˛Cˇg, the centerxı is on the real axis, the distance betweenxıandxequals1=.ı/and the radius ofDıequals 1=ı. We can supposex˛ < x˛Cˇ < xˇ. Then, the configurationD.˛; ˇ/is defined by the intersection of the union[ıDı with the vertical striphx˛; xˇi. The constant A4.r/is defined as the integral
A4.r/D 1 A.r/
“
T
d˛dˇ “
D.˛;ˇ /
yrdxdy
!
: (3.51)
Sectors. In the same vein, we consider the figure obtained with three sectors S˛; Sˇ; Sˇ˛when these sectors satisfy the following:7for anyı 2 f˛; ˇ; ˇ˛g, the sectorSı is delimited by two half lines, the real axis (with a positive orienta-tion) and another half-line, that intersect at the pointxı of the real axis. For any ı; 2 f˛; ˇ; ˇ˛g, the distance betweenxı andx equals1=.ı/. We can sup-posexˇ˛ < x˛ < xˇ; in this case, the angle of the sectorSı equals arcsinı for ı 2 fˇ˛; ˛gand equals arcsinı forı D ˇ. The configurationS.˛; ˇ/is defined by the intersection of the intersection\ıSıwith the vertical striphx˛; xˇi.
The constantA5.r/is defined as the integral A5.r/D 1
Theorem 8. (Vall´ee and Vera [45,47] 2007) When the initial density onBnFis the standard density of valuationr, the distribution of the three main output parameters involves the constantsAi.r/defined in (3.49), (3.50), (3.51), and (3.52) and satisfies the following:
1. For parameter, there is an exact formula for any valuationrand any1, P.r/Œ.z/DA1.r/.2rC3/
.2rC4/rC2 for 1
2. For parameter, there are precise estimates for any fixed valuationr > 1, whent !0,
Moreover, for any fixed valuationr >1and anyt > 0, the following inequality holds 3. For parameter, there is a precise estimate for any fixed valuationr > 1,
when u!0,
7The description is given in the case whenˇ > ˛.
P.r/Œ.z/uu!0
1
.2/A5.r/u2rC2:
Moreover, for any fixed valuation r > 1 and any u > 0, the following inequalities hold: Proof. [Sketch] If ' denotes the Euler quotient function, there are exactly '.c/
coprime pairs.a; c/witha=c21=2;C1=2. Then, the identity X
c1
'.c/
cs D .s1/
.s/ ; for <s > 2;
explains the occurrence of the function.s1/=.s/in our estimates. Consider two examples:
.a/ For1, the domain ./is made with disjoint Ford disks of radius=.2c2/.
An easy application of previous principles leads to the result.
.b/ For.t /, these same principles together with relation (3.47) entail the following inequalities
and there are several cases whent !0according to the sign ofr. Forr > 0, the Dirichlet series involved are convergent. Forr0, we consider the series
X
c1/
'.c/
crC2Cs D .sCrC1/
.sCrC2/;
(which has a pˆole atsD r), and classical estimates entail an estimate for X
For domainM.u/, the study of quadrilaterals can be performed in a similar way.
Furthermore, the height of each quadrilateral ofM.u/is .u2/, and the sum of the basesa equal 1. ThenP.r/Œ.z/ u D .u2rC2/. Furthermore, using the inclusions of (3.44) leads to the inequality.
Interpretation of the Results
We provide a first interpretation of the main results described in Theorem 8.
1. For anyy01, the probability of the eventŒbyy0is
This defines a function of the variabley07! r.y0/, whose derivative is a power function of variabley0, of the form.y0r3/. This derivative is closely related to the output densityfbrof Theorem 6 via the equality
0 den-sity, defined in (3.43), which is associated to the Haar measure2defined in Sections “Random Lattices and Relation with Eisenstein Series”.
2. The regime of the distribution function of parameterchanges when the sign of valuationr changes. There are two parts in the domain.t /: the lower part, which is the horizontal stripŒ0 =.z/.2=p
3/t2, and the upper part defined as the intersection of.t /with the horizontal stripŒ.2=p
3/t2 =.z/t . For negative values ofr, the measure of the lower part is dominant, while for positive values ofr, it is the upper part that has a dominant measure. Forr D0, there is a phase transition between the two regimes: this occurs in particular in the usual case of a uniform density.
3. In contrast, the distribution function of parameterhas always the same regime.
In particular, for negative values of valuationr, the distribution functions of the two parameters,and, are of the same form.
4. The bounds (3.53, 3.54) prove that for any u; t2Œ0; 1, the probabilitiesPŒ.z/
t ,PŒ.z/utend to 1, when the valuationrtends to1. This shows that the limit distributions ofandare associated to the Dirac measure at0.
5. It is also possible to conduct these studies in the discrete model defined in Sec-tion “Probabilistic Models for Two-Dimensions”. It is not done here, but this type of analysis will be performed in the following section.
Open question. Is it possible to describe the distribution function of parameter for > 1? Figure 3.15 [top] shows that its regime changes atD 1. This will be important for obtaining a precise estimate of the mean valueE.r/Œ as a function of rand comparing this value to experiments reported in Section “A Variation for the LLL Algorithm: The Odd-Even Algorithm”.
The corners of the fundamental domain. With Theorem 8, it is possible to com-pute the probability that an output basis lies in the corners of the fundamental domain, and to observe its evolution as a function of valuationr. This is a first step for a sharp understanding of Fig. 3.4 [right].
0 1
2
0.33333 0.4
0.25 0.2
0
12
0 1
1 2 1 3
1 4 1 5 1 6 1 7
8 2
5 3
3 7 2 8
7
Fig. 3.15 Above: the domain ./ WD fzI .z/ g. On the left, D 1(in white). On the right, the domainF.0;1/\Fo.0; 1; /for D1; 0 D2=p
3; 1 D .1C0/=2. – In the middle: the domain.t /\BC, with.t /WD fzI .z/tgand the domainM.u/\BCwith M.u/WD fzI .z/ugfor uDtD0:193.– Below: the same domains for uDtD0:12
Proposition 8. When the initial density onBnFis the standard density of valuation r, the probability for an output basis to lie on the corners of the fundamental domain is equal to
C.r/WD1A1.r/.2rC3/
.2rC4/;
whereA1.r/is defined in Section “Distribution Functions of Output Parameters:
Case of Densities with Valuations”. There are three main cases of interest for 1C.r/, namely
Œr ! 1W 3
; ŒrD0W 3 2 C3p
3 .3/
.4/; Œr! 1W r
re3=2: