Gaps between logs

0.39 MB·English

by  Jens Marklof

GAPS BETWEEN LOGS
JENS MARKLOF AND ANDREAS STRO¨MBERGSSON
Abstract. Wecalculatethelimitinggapdistributionforthefractionalpartsoflogn,
wherenrunsthroughallpositiveintegers. Byrescalingthesequence,theproofquickly
reduces to an argument used by Barra and Gaspard in the context of level spacing
statisticsforquantumgraphs. ThekeyingredientisWeylequidistributionofirrational
translationsonmulti-dimensionaltori. Ourresultsextendtologarithmswitharbitrary
base; we deduce explicit formulas when the base is transcendental or the rth root of
an integer. If the base is close to one, the gap distribution is close to the exponential
3 distribution.
1
0
2
§1. The gap distribution is a popular measure to quantify the degree of randomness
n
in a given deterministic sequence. Rudnick and Zaharescu have shown [12] that the gap
a
J distribution of the fractional parts of 2nα (n = 1,...,N), for almost all α, converges in

7 thelimitN → ∞toanexponentialdistribution—thegapdistributionofaPoissonpoint

process. (2n may be replaced by any lacunary sequence.) The same is expected to hold
]
T for the fractional parts of nkα, for any integer k ≥ 2 and α of bounded type, but this
N
remains unproven [10, 11, 8, 4]. Numerical experiments also suggest that the fractional
.
h parts of nβ may have an exponential gap distribution provided β ∈ (0, 1)∪(1,1). In the
t 2 2
a case β = 1, Elkies and McMullen [3] have calculated an explicit formula for the limiting
m 2
gap distribution, which is evidently not exponential. Remarkably, their distribution co-
[ incideswiththelimitinggapdistributionofdirectionsofpointsintheaffinelatticeZ2+a

1 for any fixed vector a ∈/ Q2 [9]. Both of these findings follow from the equidistribution
v of translates of certain curves on the homogeneous space ASL(2,Z)\ASL(2,R). In the
6
2 present paper we show that also the fractional parts of log n have a non-exponential
b
1 limiting gap distribution, which we calculate explicitly. Our derivation reduces quickly
1
to an argument used by Barra and Gaspard [1] in the context of spectral statistics of
.
1
quantum graphs. The key ingredient here is Weyl equidistribution on multi-dimensional
0
3 tori, similar in spirit to [5, Chap. 3].
1
v: §2. To state our main results, let us denote by ξn the fractional parts of logbn. We
i denote by {ξ } the ordered set of the first N elements of ξ , so that
X n,N n=1,...,N n
r (1) 0 ≤ ξ ≤ ξ ≤ ... ≤ ξ < 1.
a 1,N 2,N N,N
For purely notational reasons it will be convenient to set ξ := ξ +1, and also
N+1,N 1,N
consider the gap between the first and last element mod 1; this additional gap has of
course no effect on the existence and form of the limiting gap distribution.
For given N, the gap distribution P (s) is defined as the probability density on R ,
N ≥0
N
1 (cid:88)
(2) P (s) = δ(s−N(ξ −ξ )).
N n+1,N n,N
N
n=1
Date: 6 January 2013.
2010 Mathematics Subject Classification. 11K06, 11J71.
J.M. is supported by a Royal Society Wolfson Research Merit Award and ERC Advanced Grant
HFAKT. A.S. is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the
Knut and Alice Wallenberg Foundation.
1
2 JENS MARKLOF AND ANDREAS STRO¨MBERGSSON
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0 2.5 3.0
Figure 1. The distribution of gaps between the fractional parts of logn,
where n = 1,...,104. The piecewise continuous curve is the limit distri-
bution P(s) of Theorem 1 for b = e.
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0 2.5 3.0
Figure 2. The distribution of gaps between the fractional parts of log n
b
with base b = e1/5 = 1.221402758..., where n = 1,...,104. The blue
curve is the limit distribution P(s) of Theorem 1, the red curve is the
exponential distribution e−s.
We denote by T the set of transcendental numbers b > 1; b = e is a natural example,
cf. Figure 1. The technique we present here also works for algebraic b, but in general
leads to more intricate limit distributions. In §10 we discuss the simple case of integer
base b, and in §11 the case when the base is the rth root of an integer.
Theorem 1. Let b ∈ T . For any bounded continuous f : R → R,
≥0
(cid:90) ∞ (cid:90) ∞
(3) lim f(s)P (s)ds = f(s)P(s)ds,
N
N→∞
0 0
GAPS BETWEEN LOGS 3
where

1 s−2(F(slogb)−F(sb−1logb)) if 0 < s < 1
logb logb
(4) P(s) = − 1 s−2F(sb−1logb) if 1 < s < b
logb logb logb

0 if s > b ,
logb
∂
(5) F(a) = a (a;b−1)−(a;b−1)
∂a
and
∞
(cid:89)
(6) (a;q) = (1−aqn) (q-Pochhammer symbol).
n=0
The derivative of the Pochhammer symbol is explicitly
∞
∂ (cid:88) 1
(7) (a;q) = −(a;q) .
∂a q−j −a
j=0
Note that P(s) has jump discontinuities at s = 1 and b ; however both the left and
logb logb
right limits exist at these points, and are finite. We also note that lim P(s) = logb.
s→0+ b−1
A more elementary formulation, which is equivalent to the above, is that for every
s ≥ 0
1 (cid:12) (cid:12) (cid:90) ∞
(8) lim (cid:12){n ≤ N : N(ξ −ξ ) > s}(cid:12) = P(s(cid:48))ds(cid:48).
n+1,N n,N
N→∞ N
s
We will show in the final paragraph §12 of this paper that, in the limit b → 1, P(s)
converges to the exponential distribution e−s, cf. Figure 2.
§3. To gain a better insight into the limiting gap distribution P(s) for fixed b, let
us recall that the fractional parts of log n are not uniformly distributed mod 1 [6].
b
However, the sequence of η (n = 1,...,N) given by the fractional parts of log (n/N) =
n b
log n−log N has a limit density mod 1, and evidently the same gap distribution (to
b b
see this, recall that we have added the gap between the first and last element of the
sequence mod 1, and the ordering in the sequence remains the same). It is an easy
exercise to show that for any interval A ⊂ [0,1],
(cid:90)
1 (cid:12) (cid:12)
(9) lim (cid:12){n ≤ N : η ∈ A}(cid:12) = ρ(x)dx
n
N→∞ N
A
with density
logb
(10) ρ(x) = bx.
b−1
§4. A more refined statement of Theorem 1 is to consider the joint distribution of η
n
and the subsequent gap; define the corresponding probability density on [0,1]×R by
≥0
N
1 (cid:88)
(11) P (x,s) = δ(x−η ) δ(s−N(η −η )),
N n,N n+1,N n,N
N
n=1
where {η } is again the ordered set of the first N elements of η . The joint
n,N n=1,...,N n
distribution allows us to compare the statistics of gaps between elements near one point
x of the unit interval with the gap statistics at a second point x(cid:48).
4 JENS MARKLOF AND ANDREAS STRO¨MBERGSSON
The explicit formula for the limit distribution will involve the density
 ∂2
 (a;b−1) if 0 < a < 1
(12) R(a,b) = (b−1;b−1)δ(a−1)+ ∂a2
0 if a > 1.
The second derivative of the Pochhammer symbol is
∂2 (cid:88)∞ 1
(13) (a;q) = (a;q) .
∂a2 (q−j −a)(q−k −a)
j,k=0
j(cid:54)=k
Theorem 2. Let b ∈ T . For any bounded continuous f : [0,1]×R → R,
≥0
(cid:90) 1(cid:90) ∞ (cid:90) 1(cid:90) ∞
(14) lim f(x,s)P (x,s)dsdx = f(x,s)P(x,s)dsdx,
N
N→∞
0 0 0 0
where
(15) P(x,s) = (bx−1logb)2R(sbx−1logb,b).
§5. To unravel the previous statements further, let us consider a strictly increasing
function [0,1] → [0,1] which maps the sequence η to a new sequence η˜ which now is
j j
uniformly distributed mod 1. This function is given by (cf. [7])
(cid:90) x bx −1
(16) x (cid:55)→ ρ(y)dy = ,
b−1
0
and thus
bηn −1 bηn,N −1
(17) η˜ = , η˜ = .
n n,N
b−1 b−1
We have, explicitly, for n ∈ [Nb−k,Nb−k+1) and k ∈ Z ,
≥0
bkn−N
(18) η˜ = .
n
N(b−1)
We will see in §8 that the knowledge of the joint distribution P (x,s) for the original
N
sequence and the analogue for the rescaled and ordered sequence,
N
1 (cid:88)
˜
(19) P (x,s) = δ(x−η˜ ) δ(s−N(η˜ −η˜ ))
N n,N n+1,N n,N
N
n=1
are equivalent:
Theorem 3. Let b ∈ T . For any bounded continuous f : [0,1]×R → R,
≥0
(cid:90) 1(cid:90) ∞ (cid:90) 1(cid:90) ∞
˜ ˜
(20) lim f(x,s)P (x,s)dsdx = f(x,s)P(s)dsdx,
N
N→∞
0 0 0 0
where
(21) P˜(s) = (1−b−1)2R(cid:0)(1−b−1)s,b(cid:1).
Note that the limit distribution is independent of x. This is not a consequence of η˜
j
being uniformly distributed mod 1, but reflects the fact that the gap distribution is the
same if we restrict the sequence to an arbitrary subinterval of [0,1].
WewillfirstproveTheorem3in§6and§7, andthenretraceourstepsbysubsequently
showing in §8 that Theorem 3 implies Theorem 2, where
(22) P(x,s) = P˜(ρ(x)s)ρ(x)2 = (bx−1logb)2R(sbx−1logb,b).
GAPS BETWEEN LOGS 5
We obtain Theorem 1 from Theorem 2 by taking an x-independent test function.
§6. WenowturntotheproofofTheorem3andstudyamoregeneralsetting. Consider
a finite sequence ω ,...,ω of positive real numbers (the “frequencies”). With each ω
1 J j
we associate a real number β . We are interested in the statistical properties of the
j
multiset
J
(cid:93)
(23) S = (β +ω−1Z).
j j
j=1
Here (cid:93) denotes multiset sum, i.e. a union where we keep track of the multiplicity of
each element (this is only relevant if the sets β +ω−1Z, j = 1,...,J are not mutually
j j
disjoint). The number of elements of S that fall into the interval I = [t− L,t+ L] is
2 2
J
(24) N(t,L) = (cid:88)N (t,L), N (t,L) = (cid:12)(cid:12)I ∩(β +ω−1Z)(cid:12)(cid:12).
j j j j
j=1
Evidently N (t,L) is periodic in t with period ω−1. Furthermore
j j
(25) ω L−1 < N (t,L) ≤ ω L+1.
j j j
Therefore
J
N(t,L) (cid:88)
(26) lim = ω ,
j
L→∞ L
j=1
which means that the asymptotic density of S in R equals (cid:80) ω .
j j
For any x ∈ R we write |x| ∈ [0, 1] for the distance between x and the nearest
Z
2
integer. We also set
(27) n(x,L) = (cid:12)(cid:12)[x− L,x+ L]∩Z(cid:12)(cid:12).
2 2
The verification of the following lemma is a simple exercise.
Lemma 1. Given x ∈ R, L ≥ 0 and k ∈ Z , we have n(x,L) = k if and only if
≥0
|x− 1k| is both ≥ 1(k −L) and > 1(L−k).
Z
2 2 2
Inparticular, ifxispickedatrandomwithrespecttotheuniformprobabilitymeasure
P on [0,1], then
0
(cid:40)
1−|k −L| if L−1 < k < L+1
(28) P (n(x,L) = k) = E (k,L) :=
0 1
0 otherwise.
The following observation is due to Barra and Gaspard [1].
Theorem 4. Assume that the frequencies ω ,...,ω are linearly independent over Q.
1 J
Then, for −∞ < a < b < ∞ and k ∈ Z ,
≥0
meas({t ∈ [aT,bT] : N(t,L) = k})
(29) lim = E(k,L)
T→∞ (b−a)T
where
J
(cid:88) (cid:89) (cid:0) (cid:1)
(30) E(k,L) = E k ,ω L .
1 j j
k1+...+kJ=kj=1
6 JENS MARKLOF AND ANDREAS STRO¨MBERGSSON
In particular,
(cid:40)(cid:81)J (cid:0)1−ω L(cid:1) if L < max(ω ,...,ω )−1
(31) E(0,L) = j=1 j 1 J
0 otherwise.
The convergence in (29) is uniform over all choices of β ,...,β ∈ R (but keeping
1 J
ω ,...,ω fixed).
1 J
Proof. Decomposing[aT,bT]intointervalsoflengthω−1 andusingN(t,L) = (cid:80)J N (t,L),
1 j=1 j
we see that meas({t ∈ [aT,bT] : N(t,L) = k}) equals
(32)
n −1
(cid:88) (cid:88)b
ω−1meas({s ∈ [0,1) : N (ω−1(s+n),L) = k for j = 1,...,J})+E,
1 j 1 j
k1+...+kJ=kn=na
where n = (cid:98)ω aT(cid:99), n = (cid:98)ω bT(cid:99), and E is a real number whose absolute value
a 1 b 1
is bounded above by the measure of the symmetric difference between [aT,bT] and
[n ω−1,n ω−1]; thus |E| ≤ 2ω−1. Let us define α ∈ R/Z through the relation
a 1 b 1 1 j,n
β + ω−1Z = nω−1 + ω−1(α + Z). Then N (ω−1(s + n),L) = k holds if and only
j j 1 j j,n j 1 j
if (cid:12)(cid:12)[ω−1s − L,ω−1s + L] ∩ ω−1(α + Z)(cid:12)(cid:12) = k , and by Lemma 1 this holds if and
1 2 1 2 j j,n j
only if 2(cid:12)(cid:12)ωjs − α − 1k (cid:12)(cid:12) is both ≥ k − ω L and > ω L − k . It follows that
ω1 j,n 2 j Z j j j j
meas({t ∈ [aT,bT] : N(t,L) = k}) equals
n −1
(cid:88) (cid:88)b
(33) ω−1f (α ,α ,...,α )+E,
1 k1,...,kJ 1,n 2,n J,n
k1+...+kJ=kn=na
where f : (R/Z)J → [0,1] is defined by
k1,...,kJ
(cid:90) 1(cid:89)J (cid:16) (cid:12)ω 1 (cid:12) (cid:17)
(34) f (α ,α ,...,α ) = I 2(cid:12) js−α − k (cid:12) > |k −ω L| ds.
k1,...,kJ 1 2 J (cid:12)ω j 2 j(cid:12)Z j j
0 1
j=1
Note that each function f is continuous on (R/Z)J. It follows from our definition
k1,...,kJ
of α that α = α − ω ω−1 mod Z. In particular α is independent of n; in
j,n j,n+1 j,n j 1 1,n
fact α = α := ω β in R/Z for all n. On the other hand our assumption implies that
1,n 1 1 1
the numbers 1,ω ω−1,ω ω−1,...,ω ω−1 are linearly independent over Q, and hence by
2 1 3 1 J 1
Weyl equidistribution we have
(cid:80)nb−1 f (α ,α ,...,α ) (cid:90)
(35) n=na k1,...,kJ 1,n 2,n J,n → f (α ,α ,...,α )dα ...dα
n −n k1,...,kJ 1 2 J 2 J
b a (R/Z)J−1
as n − n → ∞. Note that the convergence here is uniform over all choices of
b a
β ,...,β ∈ R. The right hand side equals
1 J
(cid:90) (cid:90) 1(cid:89)J (cid:16) (cid:12)ω 1 (cid:12) (cid:17)
(36) I 2(cid:12) js−α − k (cid:12) > |k −ω L| ds,
(R/Z)J−1 0 (cid:12)ω1 j 2 j(cid:12)Z j j
j=1
and changing order of integration and then substituting α := x + ωjs − 1k (x ∈
j j ω1 2 j j
R/Z) we see that the expression factors as (cid:81)J E (k ,ω L). Finally noticing also that
j=1 1 j j
n −n ∼ ω (b−a)T as T → ∞, we obtain (29). (cid:3)
b a 1
We now turn to the gap distribution. We order the elements of S and label them as
(37) ... ≤ λ ≤ λ ≤ λ ≤ λ ≤ λ ≤ ....
−2 −1 0 1 2
GAPS BETWEEN LOGS 7
Theorem 5. Assume that the frequencies ω are linearly independent over Q, and that
j
ω is the largest among ω ,...,ω . Then, for −∞ < a < b < ∞, and any s ≥ 0,
1 1 J
s (cid:54)= ω−1,
1
1 (cid:90) ∞
(38) lim |{j ∈ Z : λ ∈ [aT,bT] and λ −λ > s}| = P (s(cid:48))ds(cid:48)
j j+1 j ω
T→∞ (b−a)T
s
where

J (cid:20) J (cid:21)
(cid:89)J (cid:16) ω (cid:17) (cid:88) ωhωi (cid:89) (cid:0)1−ωjs(cid:1) if 0 < s < ω1−1
(39) P (s) = ω 1− j δ(s−ω−1)+
ω 1 ω 1 h,i=1 j=1
1  h(cid:54)=i j(cid:54)=h,i
j=2 0 if s > ω−1.
1
The convergence in (38) is uniform over all choices of β ,...,β ∈ R (but keeping
1 J
ω ,...,ω fixed).
1 J
Proof. This is a corollary of Theorem 4, cf. e.g. Theorem 2.2 in [7]. The limiting gap
density is given by the formula
d2E(0,s)
(40) P (s) = .
ω ds2
(cid:3)
§7. Proof of Theorem 3. Itsufficestoprovethat(20)holdswhenf isthecharacteristic
function of a box [0,a] × [0,A] with A (cid:54)= (1 − b−1)−1, i.e. to prove that for any fixed
a ∈ [0,1] and A ≥ 0, A (cid:54)= (1−b−1)−1, we have
(cid:90) A
˜
(41) G (N) → a P(s)ds
a,A
0
as N → ∞, where
1 (cid:12)(cid:110) (cid:111)(cid:12)
(42) G (N) := (cid:12) n ∈ {1,...,N} : η˜ ∈ [0,a], N(η˜ −η˜ ) ∈ [0,A] (cid:12)
a,A N(cid:12) n,N n+1,N n,N (cid:12)
(cf., e.g., [2, Example 2.3]).
Setω = b−j(b−1)andβ(N) = β(N) = − N forj = 1,2,.... ForanygivenJ,N ∈ Z
j j b−1 ≥1
we note that by (18), {Nη˜ ,...,Nη˜ } equals the multiset sum of (β(N)+ω−1Z)∩[0,N)
1 N j
for j = 1,2,...,J and a remaining multiset of cardinality (cid:100)Nb−J(cid:101) consisting of Nη˜ for
n
n = N and all n < Nb−J. Hence if we order the elements of S := (cid:93)J (β(N) +ω−1Z)
N j=1 j
as ... ≤ λ ≤ λ ≤ λ ≤ λ ≤ λ ≤ ..., then
−2 −1 0 1 2
1 (cid:12)(cid:110) (cid:111)(cid:12)
(43) G (N) = (cid:12) j ∈ Z : λ ∈ [0,aN], λ −λ ∈ [0,A] (cid:12)+O(cid:0)N−1 +b−J(cid:1).
a,A N(cid:12) j j+1 j (cid:12)
Since b is transcendental, the frequencies ω ,...,ω are linearly independent over Q;
1 J
thus Theorem 5 applies, and we obtain, for any fixed J ∈ Z ,
≥1
(cid:90) A
(44) limsupG (N) ≤ a P (s(cid:48))ds(cid:48) +O(b−J);
a,A ω,J
N→∞ 0
(cid:90) A
liminfG (N) ≥ a P (s(cid:48))ds(cid:48) −O(b−J),
a,A ω,J
N→∞
0
8 JENS MARKLOF AND ANDREAS STRO¨MBERGSSON
where P (s) = 0 if s > ω−1 while for s < ω−1 we have
ω,J 1 1
(45) P (s) = ω (cid:89)J (cid:16)1− ωj(cid:17)δ(s−ω−1)+ d2 (cid:89)J (cid:0)1−ω s(cid:1).
ω,J 1 ω 1 ds2 j
1
j=2 j=1
We now compute that

0 if A > ω−1
 1
(cid:90) A (cid:88)J  J J
(46) P (s(cid:48))ds(cid:48) = ω − (cid:88) (cid:89)
ω,J j ω (1−ω A) if A < ω−1.
0 j=1  j i 1
j=1 i=1
(i(cid:54)=j)
Hence, letting J → ∞ in (44) and using ω = b−j(b−1), we conclude
j
(47)
 
0 if A > b−b1 (cid:90) A
˜
lim Ga,A(N) = a+a (cid:16) ∂ (cid:0) (cid:1)(cid:17) = a P(s)ds,
N→∞  ∂s (1−b−1)s;b−1 if A < b−b1 0
|s=A
where the last equality follows by a direct computation using (12) and (21). Hence (41)
holds, and Theorem 3 is proved. (cid:3)
§8. Let us now prove that Theorem 3 implies Theorem 2. (An entirely analogous
argument also shows that Theorem 2 implies Theorem 3, so that the statements of
these two theorems are in fact equivalent.) Given a bounded continuous function f :
[0,1]×R → R, we wish to prove that (14) holds. Since the sequence of probability
≥0
measures P (x,s)dsdx is tight (cf., e.g., Lemma 2.1 in [7]), a familiar approximation
N
argument shows that without loss of generality we may assume that f has compact
support, i.e. there is some B > 0 such that f(x,s) = 0 whenever s ≥ B.
Let T : [0,1] → [0,1] be the map in (16) and define T : [0,1] × R → [0,1] × R
≥0 ≥0
through T(x,s) = (T(x),T(cid:48)(x)s). This is a bijection, with inverse given by
(cid:16) s (cid:17)
(48) T−1(x,s) = T−1(x), .
T(cid:48)(T−1(x))
Now apply Theorem 3 with the test function f ◦T−1. Then in (20) we have
(cid:90) 1(cid:90) ∞ 1 (cid:88)N
(49) f ◦T−1(x,s)P˜ (x,s)dsdx = f(η ,s ),
N n,N n,N
N
0 0 n=1
T(η )−T(η )
n+1,N n,N
where s := N · . Let F be the set of those n ∈ {1,...,N} for
n,N T(cid:48)(η ) N
n,N
which η − η ≤ BbN−1. Then since T(cid:48)(x) = ρ(x) is continuous and bounded
n+1,N n,N
away from zero on [0,1] we have s = N(η − η ) + o(1) uniformly over all
n,N n+1,N n,N
n ∈ F as N → ∞, and hence, since f is uniformly continuous,
N
(50) f(η ,s ) = f(η ,N(η −η ))+o(1),
n,N n,N n,N n+1,N n,N
uniformly over all n ∈ F . On the other hand if n ∈/ F then N(η − η ) >
N N n+1,N n,N
Bb > B and also s > BbinfT(cid:48) = B, so that both sides of (50) vanish. Hence (50) in
n,N supT(cid:48)
fact holds uniformly over all n ∈ {1,...,N} as N → ∞, and it follows that the limit
in the left hand side of (14) exists, and equals the limit in (20) (with f ◦T−1). Finally
substituting (x,s) = T(x(cid:48),s(cid:48)) in the right hand side of (20) we see (via (22)) that this
limit equals the right hand side of (14), and the proof is complete.
GAPS BETWEEN LOGS 9
2.5
2.0
1.5
1.0
0.5
0.5 1.0 1.5 2.0 2.5 3.0
Figure 3. The distribution of gaps between the fractional parts of log n
b
with base b = 10, where n = 1,...,104. The piecewise continuous curve
is the limit distribution P(s) in (56).
§9. Theorem 1 is now a direct consequence of Theorem 2. As to the explicit formula
for the gap distribution,
(cid:90) 1
P(s) = P(x,s)dx
(51) 0
(cid:90) 1
= (bx−1logb)2R(sbx−1logb,b)dx.
0
With the variable substitution a = sbx−1logb, da = sbx−1(logb)2dx = (logb)adx we
have
1 (cid:90) slogb
(52) P(s) = R(a,b)ada.
s2logb
sb−1logb
Recall the definition of R(a,b) in (12). The integral over the first term in (12) yields
(b−1;b−1) if sb−1logb < 1 < slogb and 0 otherwise. For the second term, we use integra-
s2logb
tion by parts,
(cid:90) a1 ∂2 (cid:20) ∂ (cid:21)a1 (cid:90) a1 ∂
a (a;b−1)da = a (a;b−1) − (a;b−1)da
∂a2 ∂a ∂a
(53) a0 a0 a0
(cid:20) ∂ (cid:21)a1
= a (a;b−1)−(a;b−1) ,
∂a
a0
where a = min{1,sb−1logb} and a = min{1,slogb}. This yields (4).
0 1
§10. As we have noted, the technique presented here works also for algebraic b. Let
us consider briefly the simplest case of b > 1 being an integer. In this case, the multiset
sum S = (cid:93)J (β(N) + ω−1Z) considered in the proof of Theorem 3 has all its points
N j=1 j
lying in β(N) +ω−1Z, i.e. the only gap lengths appearing are 0 and ω−1 = (1−b−1)−1;
1 1
furthermore S is periodic with period ω−1 = bJ−1ω−1. One finds that the average
N J 1
multiplicity of β(N) +ω−1k in S for varying k ∈ Z equals 1−b−J. Hence following the
1 N 1−b−1
proof of Theorem 3 and taking the limit J → ∞, we conclude that for b > 1 an integer,
10 JENS MARKLOF AND ANDREAS STRO¨MBERGSSON
1.2
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0 2.5 3.0
Figure 4. The distribution of gaps between the fractional parts of log n
√ b
with base b = 10, where n = 1,...,104. The piecewise continuous curve
is the limit distribution P(s) in (62) with r = 2.
1.2
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0 2.5 3.0
Figure 5. The distribution of gaps between the fractional parts of
10log n (= log n with base b = 101/10 = 1.258925411...), where
10 b
n = 1,...,104. The blue curve is the limit distribution P(s) in (62)
with r = 10. The red curve is the exponential distribution e−s.
we have a limit result as in (20) in Theorem 3, but with the limit density being given
simply by
(54) P˜(s) = b−1δ(s)+(1−b−1)δ(cid:0)s−(1−b−1)−1(cid:1).
As to the limit of the unscaled sequence η , the first equality in (22) yields
n
logb (cid:18) b1−x(cid:19)
(55) P(x,s) = bx−1δ(s)+(logb)bx−1δ s− ,
b−1 logb