Yann Bugeaud

Lingmin Liao

Uniform Diophantine approximation related to b-ary and β-expansions

Let b ≥ 2 be an integer and v a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ξ with the property that, for every sufficiently large integer N , there exists an integer n such that 1 ≤ n ≤ N and the distance between b n ξ and its nearest integer is at most equal to b -vN . We further solve the same question when replacing b n ξ by T n β ξ, where T β denotes the classical β-transformation.

Introduction and results

Throughout this text, • stands for the distance to the nearest integer. Let ξ be an irrational real number. The well-known Dirichlet Theorem asserts that for every real number X ≥ 1, there exists an integer x with 1 ≤ x ≤ X and xξ < X -1 .

(

This is a uniform statement in the sense that (1.1) has a solution for every sufficiently large real number X (as opposed to 'for arbitrarily large real numbers X'). Following the notation introduced in [START_REF] Bugeaud | Exponents of Diophantine Approximation and Sturmian Continued Fractions[END_REF], we denote by ŵ1 (ξ) the supremum of the real numbers w such that, for any sufficiently large real number X, the inequality xξ < X -w has an integer solution x with 1 ≤ x ≤ X. The Dirichlet Theorem implies that ŵ1 (ξ) ≥ 1.

In 1926, Khintchine [START_REF] Ya | Über eine Klasse linearer diophantischer Approximationen[END_REF] established that, in fact, ŵ1 (ξ) = 1 always holds. To see this, let (p ℓ /q ℓ ) ℓ≥1 denote the sequence of convergents to ξ. If ŵ1 (ξ) > 1, then there exists a positive real number ε such that, for every sufficiently large integer ℓ, the inequality qξ < (q ℓ -1) -1-ε has an integer solution q with 1 ≤ q < q ℓ . However, it follows from the theory of continued fractions that qξ ≥ q ℓ-1 ξ ≥ 1/(2q ℓ ). This gives q ε/2 ℓ < 2, thus we have reached a contradiction. Consequently, the set of values taken by the exponent of Diophantine approximation ŵ1 is easy to determine.

In the present paper, we first restrict our attention to approximation by rational numbers whose denominator is a power of some given integer b ≥ 2 and we consider the following exponents of approximation.

2010 Mathematics Subject Classification : 37B10, 11J04, 11A63, 11K55. Definition 1.1. Let ξ be an irrational real number. Let b be an integer with b ≥ 2. We denote by v b (ξ) the supremum of the real numbers v for which the equation

b n ξ < (b n ) -v
has infinitely many solutions in positive integers n. We denote by vb (ξ) the supremum of the real numbers v for which, for every sufficiently large integer N , the equation

b n ξ < (b N ) -v
has a solution n with 1 ≤ n ≤ N .

The exponents v b have already been introduced in [START_REF] Amou | Expansions in integer bases and exponents of Diophantine approximation[END_REF]; see also Chapter 7 of [START_REF] Bugeaud | Distribution modulo one and Diophantine approximation[END_REF]. Roughly speaking, the quantity v b (ξ) measures the maximal lengths of blocks of digits 0 (or of digits b -1) in the b-ary expansion of ξ. The exponents vb are, like ŵ1 , exponents of uniform approximation. Although they occur rather naturally, they do not seem to have been studied until now.

Alternatively, we may consider the quantity ℓ n (ξ), defined as the maximal length of a block of digits 0 or a block of digits b -1 among the n first b-ary digits of ξ. Let v and v be positive real numbers. We have v b (ξ) ≥ v (resp., vb (ξ) ≥ v) if, and only if, there are arbitrarily large integers n (resp., for every sufficiently large integer n) such that ℓ n (ξ)/n ≥ v/(1 + v) (resp., ℓ n (ξ)/n ≥ v/(1 + v)).

It follows immediately from the above mentioned result of Khintchine that vb (ξ) ≤ 1. An easy covering argument shows that the set {ξ ∈ R : v b (ξ) = 0} has full Lebesgue measure. Since 0 ≤ vb (ξ) ≤ v b (ξ), (1.2) almost all real numbers ξ (with respect to the Lebesgue measure) satisfy vb (ξ) = 0. However, it is easy to construct a suitable lacunary series f (x) such that vb (f (1/b)) has a prescribed value between 0 and 1. Indeed, for any v > 0 we have

v b j≥1 b -(1+v) j = v and vb j≥1 b -(1+v) j = v v + 1 .
Observe also that

v b j≥1 b -2 j 2 = +∞ and vb j≥1 b -2 j 2 = 1.
Hence, for every v in R >0 ∪ {+∞} and every real number v in [0, 1], the sets

V b (v) := {ξ ∈ R : v b (ξ) = v} and V b (v) := {ξ ∈ R : vb (ξ) = v}
are non-empty. Again, an easy covering argument yields that dim V b (+∞) = 0, (

where dim stands for the Hausdorff dimension. Let v be in

[0, 1]. It follows from (1.2) that dim V b (v) ≤ dim{ξ ∈ R : vb (ξ) ≥ v} ≤ dim{ξ ∈ R : v b (ξ) ≥ v}. Combined with dim{ξ ∈ R : v b (ξ) ≥ v} = 1 1 + v , (1.4)
which follows from a general result of Borosh and Frankel [START_REF] Borosh | A generalization of Jarník's theorem on Diophantine approximations[END_REF], this gives

dim V b (v) ≤ 1 1 + v .
Note that (1.4) is also a special case of Theorem 5 in [START_REF] Persson | Dyadic Diophantine approximation and Katok's horseshoe approximation[END_REF] and that, furthermore, it easily follows from the mass transference principle of Beresnevich and Velani [START_REF] Beresnevich | A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures[END_REF]. Moreover, by [START_REF] Beresnevich | A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures[END_REF] or by Theorem 7 from [START_REF] Amou | Expansions in integer bases and exponents of Diophantine approximation[END_REF], we have

dim V b (v) = 1 1 + v , (1.5)
for every v ≥ 0. Our first result gives the Hausdorff dimension of the set V b (v) for any v in [0, 1].

Theorem 1.1. Let b ≥ 2 be an integer and v be a real number in [0, 1]. Then we have

dim{ξ ∈ R : vb (ξ) ≥ v} = dim{ξ ∈ R : vb (ξ) = v} = 1 - v 1 + v 2 .
(1.6) Theorem 1.1 follows from a more general statement, in which the values of both functions v b and vb are prescribed. Theorem 1.2. Let b ≥ 2 be an integer. Let θ and v be positive real numbers with v < 1.

If θ < 1/(1 -v), then the set {ξ ∈ R : vb (ξ) ≥ v} ∩ {ξ ∈ R : v b (ξ) = θv} is empty. Otherwise, we have dim({ξ ∈ R : vb (ξ) = v} ∩ {ξ ∈ R : v b (ξ) = θv}) = θ -1 -θv (1 + θv)(θ -1)
.

(1.7)

Furthermore, dim{ξ ∈ R : vb (ξ) = 1} = 0. (1.8)
A key observation in the proof of Theorem 1.2 is the fact that the right hand side inequality of (1.2) can be considerably improved. Namely, we show in Subsection 2.1 that

v b (ξ) is infinite when vb (ξ) = 1 (1.9) and v b (ξ) ≥ vb (ξ)/(1 -vb (ξ)) when vb (ξ) < 1.
The latter inequality immediately implies the first statement of Theorem 1.2. Furthermore, the combination of (1.3) and (1.9) gives (1.8).

A rapid calculation shows that the right hand side of (1.7) is a continuous function of the parameter θ on the interval [1/(1 -v), +∞), reaching its maximum at the point θ 0 := 2/(1-v) and only at that point. This maximum is precisely equal to (1-v) 2 /(1+v) 2 , namely the right hand side of (1.6). This essentially shows that Theorem 1.2 implies Theorem 1.1 (a complete argument is given at the end of Subsection 2.1).

We remark that Theorem 1.2 allows us to reprove (1.5). To see this, write v = θv,

then v = v/θ and (1.7) becomes dim({ξ ∈ R : vb (ξ) = v/θ} ∩ {ξ ∈ R : v b (ξ) = v}) = θ -1 -v (1 + v)(θ -1) = 1 1 + v 1 - v θ -1 .
Letting θ tend to infinity, we see that dim

V b (v) ≥ 1/(1 + v).
The reverse inequality can easily be obtained by using the natural covering.

Beside b-ary expansions, we can as well consider β-expansions. For β > 1, let T β be the β-transformation defined on [0, 1] by

T β (x) := βx -⌊βx⌋,
where ⌊•⌋ denotes the integer part function. We assume that the reader is familiar with the classical results on β-expansions. Some useful facts are recalled in Section 3.

For a real number β > 1, we define in a similar way the functions v β and vβ .

Definition 1.3. Let β > 1 be a real number. Let x ∈ [0, 1]. We denote by v β (x) the supremum of the real numbers v for which the equation

T n β x < (β n ) -v
has infinitely many solutions in positive integers n. We denote by vβ (x) the supremum of the real numbers v for which, for every sufficiently large integer N , the equation

T n β x < (β N ) -v
has a solution n with 1 ≤ n ≤ N .

Observe that Definitions 1.1 and 1.3 do not coincide when β is an integer at least equal to 2. However, this should not cause any trouble and we may consider that the following result, established by Shen and Wang [START_REF] Shen | Shrinking target problems for beta-dynamical system[END_REF], extends (1.4) to β-expansions.

Theorem SW. Let β > 1 be a real number and v be a positive real number. Then,

dim{x ∈ [0, 1] : v β (x) ≥ v} = 1 1 + v .
We establish the following analogues of Theorems 1.1 and 1.2 for β-expansions.

Theorem 1.4. Let β > 1 be a real number. Let θ and v be positive real numbers with

v < 1. If θ < 1/(1 -v), then the set {x ∈ [0, 1] : vβ (x) ≥ v} ∩ {x ∈ [0, 1] : v β (x) = θv} is empty. Otherwise, we have dim({x ∈ [0, 1] : vβ (x) = v} ∩ {x ∈ [0, 1] : v β (x) = θv}) = θ -1 -θv (1 + θv)(θ -1)
.

Furthermore, dim({x ∈ [0, 1] : vβ (x) = 1}) = 0.
In the same way as Theorem 1.2 implies Theorem 1.1, the next statement follows from Theorem 1.4.

Theorem 1.5. Let β > 1 be a real number and v be a real number in

[0, 1]. Then we have dim{x ∈ [0, 1] : vβ (x) ≥ v} = dim{x ∈ [0, 1] : vβ (x) = v} = 1 - v 1 + v 2 .
Persson and Schmeling [START_REF] Persson | Dyadic Diophantine approximation and Katok's horseshoe approximation[END_REF] have adopted another point of view, by considering the β-expansions of 1 and letting β vary.

Theorem PS. Let β 0 , β 1 and v be real numbers with 1 < β 0 < β 1 < 2 and v > 0. Then,

dim{β ∈ (β 0 , β 1 ) : v β (1) ≥ v} = 1 1 + v .
The assumption β 1 < 2 in Theorem PS can easily be removed; see [START_REF] Li | Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions[END_REF]. Applying some of the ideas from [START_REF] Persson | Dyadic Diophantine approximation and Katok's horseshoe approximation[END_REF], we obtain the following theorem.

Theorem 1.6. Let θ and v be positive real numbers with v < 1.

If θ < 1/(1 -v), then the set {β > 1 : vβ (1) ≥ v} ∩ {β > 1 : v β (1) = θv} is empty. Otherwise, we have dim({β > 1 : vβ (1) = v} ∩ {β > 1 : v β (1) = θv}) = θ -1 -θv (1 + θv)(θ -1)
.

Furthermore, Our paper is organized as follows. Theorem 1.2 is proved in Section 2. We recall classical results from the theory of β-expansion in Section 3 and establish Theorems 1.4 and 1.6 in Sections 4 and 5, respectively. Diophantine approximation on Cantor sets is briefly discussed in Section 6.

dim{β > 1 : vβ (1) ≥ v} = 1 - v 1 + v
Throughout this text, we denote by |I| the length of the interval I and we use ♯ to denote the cardinality of a finite set. Let v be a real number with 0 < v ≤ 1. We wish to bound from above the dimension of {ξ ∈ R : vb (ξ) ≥ v}.

By (1.3), it is sufficient to consider the set

{ξ ∈ R : vb (ξ) ≥ v} ∩ {ξ ∈ R : v b (ξ) < +∞}.
Let ξ be an irrational real number. Throughout this section, in view of the preceding observations, we assume that

0 < vb (ξ) ≤ v b (ξ) < +∞. Let ξ := ⌊ξ⌋ + j≥1 a j b j
denote the b-ary expansion of ξ. It is understood that the digits a 1 , a 2 , . . . all belong to the set {0, 1, . . . , b -1}. Define the increasing sequences (n ′ k ) k≥1 and (m ′ k ) k≥1 as follows: for k ≥ 1, we have either

a n ′ k > 0, a n ′ k +1 = . . . = a m ′ k -1 = 0, a m ′ k > 0 or a n ′ k < b -1, a n ′ k +1 = . . . = a m ′ k -1 = b -1, a m ′ k < b -1. Furthermore, for every j such that a j = 0 or b -1, there exists an index k satisfying n ′ k < j < m ′ k . Since v b (ξ) is positive, we get lim sup k→+∞ (m ′ k -n ′ k ) = +∞. (2.1)
Now, we take the maximal subsequences (n k ) k≥1 and (m k ) k≥1 of (n ′ k ) k≥1 and (m ′ k ) k≥1 , respectively, in such a way that the sequence (m k -n k ) k≥1 is non-decreasing. More precisely, take

n 1 = n ′ 1 and m 1 = m ′ 1 . Let k ≥ 1 be such that n k = n ′ j k and m k = m ′ j k
have been defined. Set

j k+1 := min{j > j k : m ′ j -n ′ j ≥ m k -n k }.
Then, define

n k+1 = n ′ j k+1
and m k+1 = m ′ j k+1 . Observe that, by (2.1), the sequence (j k ) k≥1 is well defined. Furthermore, m k -n k tends to infinity as k tends to infinity.

Note that

b n k -m k < b n k ξ < b n k -m k +1 .
By construction, we have

v b (ξ) = lim sup k→+∞ m k -n k n k = lim sup k→+∞ m k n k -1 (2.2) and vb (ξ) = lim inf k→+∞ m k -n k n k+1 ≤ lim inf k→+∞ m k -n k m k = 1 -lim sup k→+∞ n k m k . (2.3) Since lim sup k→+∞ n k m k • lim sup k→+∞ m k n k ≥ 1, we derive from (2.2) and (2.3) that vb (ξ) ≤ 1 - 1 1 + v b (ξ) = v b (ξ) 1 + v b (ξ)
.

Noticing that vb (ξ) < 1 since v b (ξ) is assumed to be finite, we have proved that

v b (ξ) ≥ vb (ξ) 1 -vb (ξ) . (2.4)
Let ξ be a real number with vb (ξ) ≥ v. Take a subsequence (k j ) j≥1 along which the supremum of (2.2) is obtained. For simplicity, we still write (n k ) k≥1 , (m k ) k≥1 for the subsequences (n k j ) j≥1 , (m k j ) j≥1 . We remark that when passing to the subsequence, the first equality in (2.3) becomes an inequality.

Let ε be a real number with 0

< ε < v b (ξ)/2. Observe that for k large enough we have (v b (ξ) -ε)n k ≤ m k -n k ≤ (v b (ξ) + ε)n k (2.5) and m k -n k ≥ n k+1 (v -ε). (2.6)
The last inequality means that the length of the block of 0 (or b -1) starting at index

n k + 1 is at least equal to n k+1 (v -ε).
The combination of the second inequality of (2.5) and (2.6) gives

(v b (ξ) + ε)n k ≥ (v -ε)n k+1 .
Consequently, there exist an integer n ′ and a positive real number ε ′ such that the sum of all the lengths of the blocks of 0 or b -1 in the prefix of length n k of the infinite sequence a 1 a 2 . . . is, for k large enough, at least equal to

(v -ε)n k 1 + v -ε v b (ξ) + ε + v -ε v b (ξ) + ε 2 + . . . -n ′ = n k (v -ε)(v b (ξ) + ε) v b (ξ) -v + 2ε -n ′ ≥ n k vv b (ξ) v b (ξ) - v -ε ′ = n k θv θ -1 -ε ′ , (2.7)
where the parameter θ is defined by

v b (ξ) = θv.
Note that, by (2.4), we must have

θ ≥ 1 1 - v . (2.8)
By the first inequality of (2.5), we have

m k ≥ (1 + v b (ξ) -ε)n k ≥ (1 + v b (ξ) -ε)m k-1 ,
for k large enough. Thus (m k ) k≥1 increases at least exponentially. Since n k ≥ m k-1 for k ≥ 2, the sequence (n k ) k≥1 also increases at least exponentially. Consequently, there exists a positive real number C such that k ≤ C log n k , for k large enough. Now, let us construct a covering. Remind that all the integers n k , m k defined above depend on ξ. Fix v and θ with θ ≥ 1 1-v . Let (n k ) k≥1 and (m k ) k≥1 be sequences such that

lim k→∞ m k -n k n k = θv, and lim inf k→∞ m k -n k n k+1 ≥ v.
For fixed k, we collect all those ξ with vb (ξ) = v and v b (ξ) = θv whose b-ary expansions have blocks of 0 (or of b -1) between n k and m k .

By the precedent analysis, there are at most C log n k blocks of 0 (or of b -1) in the prefix of length n k of the infinite sequence a 1 a 2 . . .. Since there are, obviously, at most n k possible choices for their first index, we have in total at most

2 C log n k n C log n k k = (2n k ) C log n k possible choices.
For each of these choices, it follows from (2.7) that at least n k (θv/(θ -1) -ε ′ ) digits are prescribed (and are equal to 0 or b -1). Consequently, defining the positive real number ε ′′ by the next equality, at most

n k -n k θv θ -1 -ε ′ + 1 = n k (1 + ε ′′ ) • θ -1 -θv θ -1 digits in the
-m k ≤ b -(1+θv)(1-ε ′′ )n k ,
for k large enough. We have shown that the set of those ξ corresponding to (n k ) k≥1 , (m k ) k≥1 is covered by 1) .

(2n k ) C log n k b n k (1+ε ′′ )(θ-1-θv)/(θ-
intervals of length at most b -(1+θv)(1-ε ′′ )n k .
Then, a standard covering argument shows that we have to consider the series

N≥1 (2N ) C log N b N(1+ε ′′ )(θ-1-θv)/(θ-1) b -(1+θv)(1-ε ′′ )Ns .
(2.9)

The critical exponent s 0 such that (2.9) converges if s > s 0 and diverges if s < s 0 is given by

s 0 = 1 + ε ′′ 1 -ε ′′ • θ -1 -θv (1 + θv)(θ -1)
.

It then follows that dim({ξ ∈ R : vb (ξ) ≥ v} ∩ {ξ ∈ R : v b (ξ) = θv}) ≤ θ -1 -θv (1 + θv)(θ -1)
.

(2.10)

Actually, we have proved that, for every θ ≥ (1 -v) -1 and for every sufficiently small positive number δ, we have

dim({ξ ∈ R : vb (ξ) ≥ v} ∩ {ξ ∈ R : θ ≤ v b (ξ) ≤ θ + δ}) ≤ θ -1 -θv (1 + θv)(θ -1) + 5δv.
Regarding the right-hand side of (2.10) as a function of θ and taking (2.8) into account, a short calculation shows that the maximum is attained for θ = 2/(1 -v), giving, for any positive ε, that

dim{ξ ∈ R : vb (ξ) ≥ v} ≤ 1 + ε 1 -ε (1 -v) 2 (1 + v) 2 .
We have established the required upper bound.

Lower bound.

To obtain the lower bound, we construct a suitable Cantor type set. Let v be in (0, 1). Let θ be a real number with θ

≥ 1 1-v . Choose two sequences (m k ) k≥1 and (n k ) k≥1 such that n k < m k < n k+1 for k ≥ 1, and such that (m k -n k ) k≥1 is non-decreasing. Furthermore, we assume that lim k→+∞ m k -n k n k+1 = v, (2.11)
and lim

k→+∞ m k -n k n k = θv.
(2.12)

An easy way to construct such sequences is to start with

n ′ k = θ k , m ′ k = ⌊(θv + 1)n ′ k ⌋ ,
and then to make a small adjustment to guarantee that (m k -n k ) k≥1 is non-decreasing. We consider the set of real numbers ξ in (0, 1) whose b-ary expansion ξ = j≥1 a j b j satisfies, for k ≥ 1,

a n k = 1, a n k +1 = . . . = a m k -1 = 0, a m k = 1,
and

a m k +(m k -n k ) = a m k +2(m k -n k ) = • • • = a m k +t k (m k -n k ) = 1,
where t k is the largest integer such that

m k + t k (m k -n k ) < n k+1 . Observe that, since t k < n k+1 -m k m k -n k ≤ 2 v ,
for k large enough, the sequence (t k ) k≥1 is bounded.

We check that the maximal length of blocks of zeros in the prefix of length n k+1 of the infinite sequence a 1 a 2 . . . is equal to m k -n k -1. Thus, we deduce that vb (ξ) = v and v b (ξ) = θv.

Actually, the above two equalities might not be true if b = 2. However, they will be valid if we take the block 10 in place of 1 in the definition of a m k +t(m k -n k ) for t = 1, . . . , t k . The following proof will be almost the same. So, for simplicity, we assume that b ≥ 3 and leave to the reader the slight change to deal with the case b = 2 (or the reader may look at Subsection 4.2).

Our Cantor type subset E θ,v consists precisely of the real numbers in (0, 1) whose b-ary expansion has the above property. We will now estimate the Hausdorff dimension of E θ,v from below.

Let n be a large positive integer. For a 1 , . . . , a n in {0, 1, . . . , b -1}, denote by I n (a 1 , . . . , a n ) the interval composed of the real numbers in (0, 1) whose b-ary expansion starts with a 1 . . . a n . Define a Bernoulli measure µ on E θ,v as follows. We distribute the mass uniformly. For k ≥ 1, set

δ k := m k -n k -1 and u k := m k + t k (m k -n k ). If there exists k ≥ 2 such that n k ≤ n ≤ m k , then define µ(I n (a 1 , . . . , a n )) = b -(n 1 -1+ k-1 j=1 (t j δ j +n j+1 -u j -1)) = b -(n k -1- k-1 j=1 (m j -n j +1+t j )) .
Observe that we have µ(

I n k ) = µ(I n k +1 ) = . . . = µ(I m k ).
If there exists k ≥ 2 such that m k < n < n k+1 , then define

µ(I n (a 1 , . . . , a n )) = b -(n 1 -1+ k-1 j=1 (t j δ j +n j+1 -u j -1)+tδ k +n-(m k +tδ k )) = b -n+ k-1 j=1 (m j -n j +1+t j )+m k -n k +1+t ,
where t is the largest integer such that m k + t(m k -n k ) ≤ n. It is routine to check that µ is well defined on E θ,v . Now we calculate the local dimension of µ at x ∈ E θ,v , i.e., lim inf

r→0 log µ(B(x, r)) log r ,
where B(x, r) stands for the ball centered at x with radius r. To this end, we first calculate the same lower limit for the basic intervals, and we prove that

lim inf n→∞ log µ(I n ) log |I n | = θ -1 -θv (θ -1)(θv + 1)
.

(2.13)

Since the lengths of basic intervals decrease 'regularly', the above limit (2.13) is the same as the local dimension of µ at x ∈ E θ,v . The details are left to the reader. Finally, the required lower bound follows from the mass distribution principle; see [START_REF] Falconer | Techniques in fractal geometry[END_REF], page 26.

Let us finish the proof by showing (2.13). First, we check that if n = m k , then lim inf

k→∞ log µ(I m k ) log |I m k | = lim inf k→∞ n k -1 - k-1 j=1 (m j -n j + 1 + t j ) m k = lim inf k→∞ n 1 -1 + k-1 j=1 (n j+1 -m j + 1 + t j ) m k .
Recalling that (t k ) k≥1 is bounded and that (m k ) k≥1 grows exponentially fast in terms of k, we have lim inf

k→∞ log µ(I m k ) log |I m k | = lim inf k→∞ k-1 j=1 (n j+1 -m j ) m k .
By (2.11) and (2.12), we see that

lim k→∞ m k n k = θv + 1, lim k→∞ m k+1 m k = θ, and lim k→∞ n k+1 m k = θ θv + 1 .
Thus, by the Stolz-Cesàro Theorem,

lim k→∞ k-1 j=1 (n j+1 -m j ) m k = lim k→∞ n k+1 -m k m k+1 -m k = lim k→∞ n k+1 m k -1 m k+1 m k -1 = θ -1 -θv (θ -1)(θv + 1)
.

Hence,

lim inf k→∞ log µ(I m k ) log |I m k | = θ -1 -θv (θ -1)(θv + 1)
.

Let n be a large positive integer. If there exists k ≥ 2 such that

n k ≤ n ≤ m k , then log µ(I n ) log |I n | ≥ log µ(I n ) log |I m k | = log µ(I m k ) log |I m k | .
If there exists k ≥ 2 such that m k < n < n k+1 , write n = m k + t(m k -n k ) + ℓ, where t, ℓ are integers with 0 ≤ t ≤ t k and 0 ≤ ℓ < m k -n k . Then we have

µ(I n ) = µ(I m k ) • b -(tδ k +ℓ) and |I n | = |I m k | • b -(t(m k -n k )+ℓ) .
Since 0 ≤ t ≤ t k and (t k ) k≥1 is bounded, for n large enough,

-log µ(I n ) -log |I n | = -log µ(I m k ) + tδ k + ℓ -log |I m k | + t(m k -n k ) + ℓ ≥ -log µ(I m k ) -log |I m k | ,
where we have used the fact that

a + x b + x ≥ a b
, for all 0 < a ≤ b, x ≥ 0.

We have established (2.13).

Classical results on β-expansions

Throughout this section, β denotes a real number greater than 1 and ⌊β⌉ is equal to β -1 if β is an integer and to ⌊β⌋ otherwise. The notion of β-expansion was introduced by Rényi [START_REF] Rényi | Representations for real numbers and their ergodic properties[END_REF] in 1957. We denote by T β the transformation defined on [0, 1] by T β (x) = {βx}, where {•} denotes the fractional part function.

Definition 3.1. The expansion of a number x in [0, 1] to base β, also called the βexpansion of x, is the sequence

(ε n ) n≥1 = (ε n (x, β)) n≥1 of integers from {0, 1, . . . , ⌊β⌉} such that x = ε 1 β + ε 2 β 2 + . . . + ε n β n + . . . ,
and, unless x = 1 and β is an integer, defined by one of the following equivalent properties:

k>n ε k β k < 1 β n , for all n ≥ 0; ε 1 = ⌊βx⌋, ε 2 = ⌊β{βx}⌋, ε 3 = ⌊β{β{βx}}⌋, . . . ε n = ⌊βT n-1 β (x)⌋, for all n ≥ 1.
We then write

d β (x) = ε 1 ε 2 . . . ε n . . . .
For x < 1, the β-expansion coincides with the representation of x computed by the 'greedy algorithm'. If β is an integer, then the digits ε i of x lie in the set {0, 1, . . . , β -1} and d β (x) corresponds, for x = 1, to the usual β-ary expansion of x. If the β-expansion

d β (1) = ε 1 (1, β)ε 2 (1, β) . . . ε n (1, β) . . .
(ε * 1 (β), ε * 2 (β), . . . , ε * n (β), . . .) := (ε * 1 (1, β), ε * 2 (1, β), . . . , (ε m (1, β) -1)) ∞ ,
where (w) ∞ stands for the periodic sequence (w, w, w, . . .).

We endow the set {0, 1, . . . , ⌊β⌉} Z ≥1 with the product topology and the one-sided shift operator σ defined by σ((s n ) n≥1 ) = (s n+1 ) n≥1 , for any infinite sequence (s n ) n≥1 in {0, 1, . . . , ⌊β⌉} Z ≥1 .

The lexicographic order on {0, 1, . . . , ⌊β⌉} Z ≥1 , denoted by < lex , is defined as follows: we write w = (w 1 , w 2 , . . .) < lex w ′ = (w ′ 1 , w ′ 2 , . . .) if there exists k ≥ 1 such that for all j < k we have w j = w ′ j , but w k < w ′ k . We use the notation w ≤ lex w ′ if w < lex w ′ or w = w ′ . Definition 3.2. The closure of the set of all β-expansions of x in [0, 1] is called the β-shift and denoted by Σ β .

Parry [START_REF] Parry | On the β-expansions of real numbers[END_REF] proved that the β-shift Σ β is fully determined by d β (1).

σ k (s) ≤ lex (ε 1 , . . . , ε m-1 , (ε m -1)) ∞ , for k ≥ 1. If d β (1)
σ k (ε 1 . . . ε m ) ≤ lex d β (1), for k = 0, 1, . . . , m -1, respectively, if σ k (ε 1 ε 2 . . .) ≤ lex d β (1), for k ≥ 0. An infinite sequence (ε 1 , ε 2 , . . .) is self-admissible if σ k (ε 1 ε 2 . . .) ≤ lex (ε 1 ε 2 . . .), for k ≥ 0.
Denote by Σ n β the set of all β-admissible blocks of length n. Then, its cardinality satisfies (Rényi [14], formula 4.9)

β n ≤ ♯Σ n β ≤ β n+1 β -1 . (3.1)
For any (ε 1 , . . . , ε n ) ∈ Σ n β , call

I n (ε 1 , . . . , ε n ) := {x ∈ [0, 1] : d β (x) starts with ε 1 . . . ε n }
an n-th order basic interval (with respect to the base β). Denote by I n (x) the n-th order basic interval containing x. We remark that the basic intervals are also called cylinders by some authors.

The next theorem was proved by Parry [START_REF] Parry | On the β-expansions of real numbers[END_REF].

Theorem 3.5. A sequence of digits (ε 1 , ε 2 , . . .) is the β-expansion of 1 for some β > 1 if and only if it is self-admissible. Now, we estimate the length of the basic intervals. We will use the notion of "full cylinder" introduced by Fan and Wang [START_REF] Fan | On the lengths of basic intervals in beta expansions[END_REF].

Definition 3.6. Let (ε 1 , . . . , ε n ) ∈ Σ n β be a β-admissible block in Σ β . A basic interval I n (ε 1 , . . . , ε n ) is called full if it is of length β -
|I n+m (w, w ′ )| = |I n (w)| • |I m (w ′ )| = β -n |I m (w ′ )|.
The following approximation of β-shift is very useful. For β > 1, let (ε * k (β)) k≥1 denote the infinite β-expansion of 1. Let β N be the unique real number which satisfies the equation

1 = ε * 1 (β) z + • • • + ε * N (β) z N .
Then we have β N < β and the sequence (β N ) N≥1 increases and converges to β when N tends to infinity. Furthermore, the subshift of finite type Σ β N is a subset of the β-shift Σ β . The subsets Σ β N are increasing and converge to Σ β .

a n ′ k > 0, a n ′ k +1 = . . . = a m ′ k -1 = 0, a m ′ k > 0,
and, furthermore, for every j such that a j = 0, there exists an index k satisfying n ′ k < j < m ′ k . Then take the maximal subsequences (n k ) k≥1 and (m k ) k≥1 of (n ′ k ) k≥1 and (m ′ k ) k≥1 , respectively, in such a way that the sequence (m k -n k ) k≥1 is non-decreasing. Observe that, since v β (x) > 0, this sequence tends to infinity with k. Similarly, notice that

β n k -m k < T n k β x < β n k -m k +1 .
We also have

v β (x) = lim sup k→+∞ m k -n k n k , vβ (x) ≤ lim inf k→+∞ m k -n k n k+1 ,
and the relations

vβ (x) ≤ v β (x) 1 + v β (x) and v β (x) ≥ vβ (x) 1 -vβ (x)
.

The last inequality means that v β (x) is infinite when vβ (x) = 1. Combined with Theorem SW, it implies the last assertion of Theorem 1.4. Let v be in (0, 1). Let θ be the real number defined by

v β (x) = θv. Then θ ≥ 1 1 - v .
Arguing as in the proof of Theorem 1.2, we see that there exists a positive real number ε such that the sum of all the lengths of the blocks of zeros in the prefix of length n k of the infinite sequence a 1 a 2 . . . is at least equal to

n k θv θ -1 -ε , for k ≥ 1.
Now we study the possible choices of digits among a 1 , . . . , a m k as we did in Subsection 2.1. Note that there are k blocks of digits which are 'free'. Denote their lengths by ℓ 1 , . . . , ℓ k . Then we have ℓ j = n j -m j-1 and

k j=1 ℓ j ≤ n k -n k θv θ -1 -ε = n k (1 + ε ′ )(θ -1 -θv)/(θ -1),
with ε ′ > 0 being a small real number. By (3.1), there are at most

β β -1 • β ℓ j
ways to choose the block of length ℓ j . Thus, for the k blocks, we have in total

β β -1 k • β k j=1 ℓ j ≤ β β -1 k • β n k (1+ε ′ )(θ-1-θv)/(θ-1)

choices.

As in Subsection 2.1, there exists a real number C > 1 such that k is less than C log n k , and there are at most n k possible choices for the first index of the k blocks. Thus, we have at most n C log n k k possible choices for the k blocks. We get that, for k sufficiently large, the set of real numbers x with the above properties is contained in a union of no more than

βn k β -1 C log n k β n k (1+ε ′ )(θ-1-θv)/(θ-1)
basic intervals of order m k , each of them being of length at most β -m k . Furthermore, there are infinitely many indices k such that

β -m k ≤ β -(1+θv)(1-ε ′ )n k .
Arguing as in Subsection 2.1, we end up with the series

N≥1 N C log N β N(1+ε ′ )(θ-1-θv)/(θ-1) β -(1+θv)(1-ε ′ )Ns . (4.1)
The critical exponent s 0 such that (4.1) converges if s > s 0 and diverges if s < s 0 is given by

s 0 = 1 + ε ′ 1 -ε ′ • θ -1 -θv (1 + θv)(θ -1)
.

We then get that dim({x ∈ (0, 1) : vβ (x) ≥ v} ∩ {x ∈ (0, 1) :

v β (x) = θv}) ≤ θ -1 -θv (1 + θv)(θ -1)
.

For the rest of the proof, we argue exactly as at the end of Subsection 2.1. We omit the details.

Lower bound.

Let v be in (0, 1). Let θ be a real number with θ ≥ 1 1-v . We construct a subset E θ,v whose elements x satisfy vβ (x) = v and v β (x) = θv.

We will suitably modify the construction we performed in Subsection 2.2 when dealing with b-ary expansions.

Let N be a positive integer. Let β N be the real number defined from the infinite βexpansion of 1 as explained at the end of Section 3. As in Subsection 2.2, let (m k ) k≥1 and (n k ) k≥1 be sequences of positive integers with n k < m k < n k+1 for k ≥ 1, such that (m kn k ) k≥1 is non-decreasing and (2.11) and (2.12) are satisfied. Start with the construction performed in Subsection 2.2 and replace the digit 1 for a n k , a m k and a m k +j(m k -n k ) , 1 ≤ j ≤ t k , by the block 0 N 10 N . Fill other places by blocks in Σ β N . Thus, we have completed the modifications and have constructed the subset E θ,v .

Since N is fixed and (t k ) k≥1 is bounded, we check that for every

x in E θ,v , vβ (x) = lim k→+∞ m k -n k -1 + 2N n k+1 + (4k + 2)N + k j=1 2N t j = lim k→+∞ m k -n k n k+1 = v, and
v β (x) = lim k→+∞ m k -n k -1 + 2N n k + (4k -2)N + k-1 j=1 2N t j = lim k→+∞ m k -n k n k = θv.
Moreover, the sequence d β (x) (in Σ β ) is, by construction, also in Σ β N .

Let n be a large positive integer. Denote by I n (a 1 , . . . , a n ) the interval composed of the real numbers in (0, 1) whose β-expansion starts with a 1 . . . a n . We will define a Bernoulli measure µ on E θ,v . We distribute the mass uniformly when we meet a block in Σ β N and keep the mass when we go through the positions where the digits are determined by the construction. Precisely, we can write down the first levels as follows.

If n < n 1 , define

µ(I n ) = 1 ♯Σ n β N . If n 1 ≤ n ≤ m 1 + 4N , then take µ(I n ) = 1 ♯Σ n 1 -1 β N .
If there exists an integer t such that 0 ≤ t ≤ t 1 and

m 1 +4N +t(m 1 -n 1 )+2N t < n ≤ min{n 2 +4N +2N t 1 , m 1 +4N +(t+1)(m 1 -n 1 )+2N t}, then set µ(I n ) = 1 ♯Σ n 1 -1 β N • 1 (♯Σ m 1 -n 1 -1 β N ) t • 1 ♯Σ n-(m 1 +4N+t(m 1 -n 1 )+2Nt) β N .
If there exists an integer t such that 0 ≤ t ≤ t 1 -1 and

m 1 + 4N + (t + 1)(m 1 -n 1 ) + 2N t < n ≤ m 1 + 4N + (t + 1)(m 1 -n 1 ) + 2N (t + 1), then set µ(I n ) = 1 ♯Σ n 1 -1 β N • 1 (♯Σ m 1 -n 1 -1 β N ) t+1
.

More generally, for k ≥ 2, set

l k := n k + (4k -4)N + k-1 j=1 2N t j , h k := m k + 4kN + k-1 j=1 2N t j . and δ k := m k -n k -1, u k := h k + t k (m k -n k ) + 2N t k . If l k ≤ n ≤ h k , define µ(I n ) = 1 ♯Σ n 1 -1 β N • 1 k-1 j=1 (♯Σ δ j β N ) t j ♯Σ l j+1 -u j -1 β N (= µ(I l k ) = µ(I h k )).
If there exists an integer t such that 0 ≤ t ≤ t k -1 and

h k + t(m k -n k ) + 2N t < n ≤ min{l k+1 , h k + (t + 1)(m k -n k ) + 2N t}, then define µ(I n ) = µ(I h k ) • 1 (♯Σ δ k β N ) t • 1 ♯Σ n-(h k +t(m k -n k )+2Nt) β N .
If there exists an integer t such that 0 ≤ t ≤ t k -1 and

h k + (t + 1)(m k -n k ) + 2N t < n ≤ h k + (t + 1)(m k -n k ) + 2N (t + 1), then define µ(I n ) = µ(I h k ) • 1 (♯Σ δ k β N ) t+1
.

By the construction and Proposition 3.7, we deduce that I h k is full and thus has length β -h k . Then, by recalling that N is fixed and (t k ) k≥1 is bounded, we deduce from (3.1) that lim inf

k→∞ log µ(I h k ) log |I h k | = lim inf k→∞ log ♯Σ n 1 -1 β N + k-1 j=1 t j log ♯Σ δ j β N + log ♯Σ l j+1 -u j -1 β N h k log β = lim k→∞ n 1 -1 + k-1 j=1 (t j δ j + l j+1 -u j -1) h k • log β N log β = lim k→∞ n 1 -1 + k-1 j=1 (l j+1 -h j -2N t j -1) h k • log β N log β = lim k→∞ k-1 j=1 (n j+1 -m j ) m k • log β N log β = θ -1 -θv (θ -1)(θv + 1) • log β N log β .
Let n be a large positive integer. If there exists k ≥ 2 such that

l k ≤ n ≤ h k , then log µ(I n ) log |I n | ≥ log µ(I n ) log |I h k | = log µ(I h k ) log |I h k | .
If there exist integers k ≥ 2 and t such that 0 ≤ t ≤ t k -1 and

h k + t(m k -n k ) + 2N t < n ≤ min{l k+1 , h k + (t + 1)(m k -n k ) + 2N t}, then, letting ℓ = n -(h k + t(m k -n k ) + 2N t), we have µ(I n ) ≤ µ(I h k ) • β -δ k t-ℓ N . Since I h k is full, then |I n | = |I h k | • |I n-h k (w ′ )|,
where w ′ is an admissible block in Σ β N of length n -h k . By Proposition 3.9,

|I n-h k (w ′ )| ≥ β -(n-h k +N) . Hence, |I n | ≥ |I h k | • β -(n-h k +N) . Notice that n -h k + N = t(m k -n k ) + 2N t + ℓ + N = δ k t + ℓ + t + N (2t + 1).
Since N is fixed and t is bounded, we argue as in Subsection 2.2 to show that, for n large enough, we have

-log µ(I n ) -log |I n | ≥ -log µ(I h k ) + (tδ k + ℓ) log β N -log |I h k | + (tδ k + ℓ + t + N (2t + 1)) log β ≥ -log µ(I h k ) -log |I h k | • η(N ),
where η(N ) < 1 and η(N ) tends to 1 as N tends to infinity. Similarly, if there exist integers k ≥ 2 and t such that 0 ≤ t ≤ t k -1 and

h k + (t + 1)(m k -n k ) + 2N t < n ≤ h k + (t + 1)(m k -n k ) + 2N (t + 1),
we also have for n large enough,

-log µ(I n ) -log |I n | ≥ -log µ(I h k ) -log |I h k | • η(N ).
So, in all cases, we have lim inf

n→∞ log µ(I n ) log |I n | ≥ θ -1 -θv (θ -1)(θv + 1) • log β N log β • η(N ).
Now, we consider a general ball B(x, r) with x a point in the Cantor-type subset E θ,v and r satisfying

|I n+1 (x)| ≤ r < |I n (x)|.
By the construction and Proposition 3.9, any n-th order basic interval I n satisfies

|I n | ≥ β -(n+N) .
Thus, the ball B(x, r) intersects at most ⌈2β N ⌉ + 2 basic intervals of order n. Noting that all n-th order basic intervals have the same measure, we deduce that

µ(B(x, r)) ≤ (⌈2β N ⌉ + 2) • µ(I n (x)).
On the other hand, by Proposition 3.9,

r ≥ |I n+1 (x)| ≥ β -(n+1+N) = β -(N+1) • β -n ≥ β -(N+1) • |I n (x)| Since N is fixed, we have lim inf r→0 log µ(B(x, r)) log r = lim inf n→∞ log µ(I n (x)) log |I n (x)| .
Finally, the lower bound follows by letting N tend to infinity.

To end this section, we remark that the last step of the proof of the lower bound, i.e., the fact that the two lower limits concerning the basic intervals and the general balls are the same, always holds in the setting of β-transformation. This conclusion was proved very recently by Bugeaud and Wang in [START_REF] Bugeaud | Distribution of full cylinders and the Diophantine properties of the orbits in β-expansions[END_REF], who called it a modified mass distribution principle.

Proof of Theorem 1.6

We follow the approach of Persson and Schmeling [START_REF] Persson | Dyadic Diophantine approximation and Katok's horseshoe approximation[END_REF]. The main idea is to take a correspondence between the β-shift and the parameter space. Then the results in the shift space can be translated to the parameter space.

Upper bound.

First we reduce the question to a small interval (β 0 , β 1 ), where 1

< β 0 < β 1 . For v ∈ [0, 1] and θ ≥ 1, set D θ,v := {β > 1 : vβ (1) = v} ∩ {β > 1 : v β (1) = θv}, and D θ,v (β 0 , β 1 ) := {β ∈ (β 0 , β 1 ) : vβ (1) = v} ∩ {β ∈ (β 0 , β 1 ) : v β (1) = θv}.
By Theorem 3.5, to every self-admissible sequence corresponds a real number β > 1. Let K β 1 be the set of all self-admissible sequences in Σ β 1 . Let π β be the natural projection from the β-shift to the unit interval [0, 1]. Then there is a one-to-one map ̺ β 1 :

π β 1 (K β 1 ) → (1, β 1 ). Let B θ,v be the subset of Σ β 1 defined by B θ,v := π -1 β 1 {x ∈ [0, 1] : vβ 1 (x) = v} ∩ {x ∈ [0, 1] : v β 1 (x) = θv} .
The Hölder exponent of the restriction of the map ̺ β 1 to the subset π β .

1 (K β 1 ∩ B θ,v ) is equal to log β 0 / log β 1 . Since D θ,v (β 0 , β 1 ) ⊂ ̺ β 1 (π β 1 (K β 1 ∩ B θ,v )) we have dim D θ,v (β 0 , β 1 ) ≤ dim ̺ β 1 (π β 1 (K β 1 ∩ B θ,v )) ≤ log β 1 log β 0 dim π β 1 (B θ,v
Letting β 1 tend to β 0 , we obtain the requested upper bound.

Lower bound.

Take β 2 such that 1 < β 0 < β 1 < β 2 and that the β 2 -expansion of 1 ends with zeros, i.e., such that the β-shift Σ β 2 is a subshift of finite type. We establish the following lemma. Lemma 5.1. For real numbers β 0 , β 1 , β 2 , v and θ such that 1 < β 0 < β 1 < β 2 , v ∈ [0, 1] and θ ≥ 1, we have

dim(̺ -1 β 2 (D θ,v (β 0 , β 1 ))) ≥ θ -1 -θv (1 + θv)(θ -1) • log β 1 log β 2 .
It follows from Lemma 5.1 and the proof of Theorem 14 of Persson and Schmeling [START_REF] Persson | Dyadic Diophantine approximation and Katok's horseshoe approximation[END_REF] that dim(D θ,v (β 0 , β 1 )) ≥ θ -1 -θv (1 + θv)(θ -1)

• log β 1 log β 2 .

Finally, letting β 2 tend to β 1 , we complete the proof of the lower bound.

Proof of Lemma 5.1.

Diophantine approximation and Cantor sets

Throughout this section, b denotes an integer at least equal to 3 and S is a subset of {0, 1, . . . , b -1} of cardinality at least two and containing at least one of the digits 0 and b -1. Let K b,S denote the set of real numbers ξ in [0, 1] which can be expressed as ξ = j≥1 a j b j , with a j ∈ K b,S for j ≥ 1. Note that for b = 3 and S = {0, 2} the set K b,S is the middle third Cantor set, which we simply denote by K. Furthermore, recall that the Hausdorff dimension of K b,S is given by dim K b,S = log ♯S log b .

As a corollary of a more general result, Levesley, Salp and Velani [START_REF] Levesley | On a problem of K. Mahler: Diophantine approximation and Cantor sets[END_REF] established that, for v ≥ 0, dim{ξ ∈ K : v 3 (ξ) ≥ v} = dim{ξ ∈ K : v 3 (ξ) = v} = log 2 log 3

• 1 1 + v . ( 6.1)
A suitable modification of the proofs of Theorems 1.1 and 1.2 allows us to extend (6.1) as follows.

• 1 - v 1 + v 2 .
The assumption that S contains at least one of the digits 0 and b -1 is necessary, since, otherwise, we trivially have v b (ξ) = 0 for every ξ in K b,S .

For additional results on Diophantine approximation on Cantor sets, the reader may consult Chapter 7 of [START_REF] Bugeaud | Distribution modulo one and Diophantine approximation[END_REF] and the references quoted therein.

Proof of Theorem 6.1. We follow step by step the proof of Theorems 1.1 and 1.2. To prove that the right hand side of (6.2) is an upper bound for the dimension, we use the same covering argument, but, instead of (2.9), we have to consider the series N≥1 (2N ) C log N (♯S) N(1+ε ′ )(θ-1-θv)/(θ-1) b -(1+θv)(1-ε ′ )Ns .

As for the lower bound, we again consider a Bernoulli measure and we distribute the mass uniformly among the elements of S. Also, if 1 does not belong to S, we cannot take a n k , a m k , . . . equal to 1 and we then choose them equal to a non-zero element of S. We omit the details.

2 ,

 2 and dim({β > 1 : vβ (1) = 1}) = 0.

2 . 2 2. 1 .

 221 Proof of Theorem 1.Upper bound.

of 1

 1 is finite, i.e., if there exists m ≥ 1 such that ε m (1, β) = 0 and ε n (1, β) = 0 for all n > m, then β is called a simple Parry number. In this case, we define the infinite β-expansion of 1 by

Theorem 3 . 3 .

 33 If d β (1) = ε 1 . . . ε m 00 . . . 0 . . . is finite with ε m = 0, then s = (s n ) n≥1 belongs to Σ β if, and only if,

π β 1 (

 1 B θ,v ) = θ -1 -θv (1 + θv)(θ -1)

Theorem 6 . 1 .

 61 Let b ≥ 3 be an integer and S a subset of {0, 1, . . . , b -1} of cardinality at least two and containing at least one of the digits 0 and b -1. Let θ and v be positive real numbers with v < 1. If θ < 1/(1 -v), then the set {ξ ∈ K b,S : vb (ξ) ≥ v} ∩ {ξ ∈ K b,S : v b (ξ) = θv} is empty. Otherwise, we have dim({ξ ∈ K b,S : vb(ξ) = v} ∩ {ξ ∈ K b,S : v b (ξ) = θv}) = log ♯S log b • θ -1 -θv (1 + θv)(θ -1). (6.2)Furthermore, dim{ξ ∈ K b,S : vb (ξ) = 1} = 0.and dim{ξ ∈ K b,S : vb (ξ) ≥ v} = dim{ξ ∈ K b,S : vb (ξ) = v} = log ♯S log b

  prefix a 1 a 2 . . . a n k , and thus in a 1 a 2 . . . a m k , are free. The set of real numbers whose b-ary expansion starts with a 1 a 2 . . . a m k defines an interval of length b -m k . By (2.5), we have b

  does not terminate with zeros only, then s = (s n ) n≥1 belongs to Σ β if, and only if, σ k (s) ≤ lex d β (1), for k ≥ 1. It follows from Theorem 3.3 that Σ β is contained in Σ β ′ if, and only if, β ≤ β ′ . Definition 3.4. A block ε 1 . . . ε m , respectively, an infinite sequence ε 1 ε 2 . . ., on the alphabet {0, 1, . . . , ⌊β⌉} is β-admissible (or, simply, admissible) if

  n .

	Proposition 3.7. ([8], Lemma 3.1) A basic interval I n (ε 1 , . . . , ε n ) is full if and only if, for any admissible block (ε ′ 1 , . . . , ε ′ m ) ∈ Σ m 1 , . . . , ε ′ m ) is also β , the concatenation (ε 1 , . . . , ε n , ε ′ admissible, i.e., in Σ n+m β .
	Proposition 3.8. ([15], Corollary 2.6) For any w ∈ Σ n β , if I n (w) is full, then for any w ′ ∈ Σ m β , we have

  Proposition 3.9. ([15], Lemma 2.7) For any w ∈ Σ n β N viewed as an element of Σ n β , we have β -(n+N) ≤ |I n (w)| ≤ β -n .

		4. Proof of Theorem 1.4
	4.1. Upper bound.
	The proof is essentially similar to that in Subsection 2.1. Let x ∈ [0, 1] be a real
	number and let	x =	a 1 β	+	a 2 β 2 + . . .
	denote its β-expansion. Assume that v β (x) is positive.
	Define the increasing sequences (n ′ k ) k≥1 and (m ′ k ) k≥1 as follows: for k ≥ 1, we have

Acknowledgements. L. Liao was partially supported by the ANR, grant 12R03191A -MUTADIS, France.

Denote by (ε * k ) k≥1 := (ε * k (β 1 )) k≥1 the infinite β 1 -expansion of 1. Take an integer N sufficiently large such that ε * N = 0 and the β 0 -expansion of 1 is smaller than ε * 1 . . . ε * N in lexicographical order. Let βN be the unique real number which satisfies the equation

Then we have β 0 < βN < β 1 , and βN tends to β 1 as N tends to infinity. Moreover, the infinite βN -expansion of 1 is given by

) by using the same construction as in Subsection 4.2. Take any sequence a ∈ Σ βN constructed in Subsection 4.2 with β = βN . Make the concatenation ε * 1 . . . ε * N 0 N a. By Lemma 5.2 from [START_REF] Li | Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions[END_REF], the sequence ε * 1 . . . ε * N 0 N a is self-admissible and thus, by Theorem 3.5, is the β-expansion of 1 for some β. By checking the lexicographic ordering, we see that β 0 < β < β 1 . We define the subset K N to be the collection of these sequences. Notice that K N is also a subset of Σ β 2 .

Now we can define a measure μ on the set π β 2 (K N ). Consider a basic interval

Since Σ β 2 is of finite type, there exists a positive real number C such that

Thus, noting that N is fixed, we can deduce, as in Section 4.2, that lim inf

Similarly as in the previous section, we have the same inequality for the general ball B(x, r), i.e., for any x in the Cantor-type set π β 2 (K N ), we have lim inf

Hence dim(̺ -1 β 2 (D θ,v (β 0 , β 1 ))) ≥ dim(π β 2 (K N )) ≥ θ -1 -θv (θ -1)(θv + 1)

• log βN log β 2 • η(N ).

Letting N tend to infinity, this proves the lemma.


Martial Bernoux
email: <martial.bernoux@ird.fr>

Giacomo Branca

; Aude Carro

Leslie Lipper

Garry Smith

Louis Bockel

Ex-ante greenhouse gas balance of agriculture and forestry development programs

Keywords: mitigation, nitrous oxide, methane, C sequestration, land use change and forestry Balanço

). Thus, EX-ACT allows for the carbon-balance appraisal of new investment programmes by ensuring an appropriate method available for donors and planning officers, project designers, and decision makers within agriculture and forestry sectors in developing countries. The tool can also help to identify the mitigation impacts of various investment project options, and thus provide an additional criterion for consideration in project selection.

Introduction

Agriculture and forestry sectors are of key concern in meeting climate change challenges, both as a source the problem and potentially as a solution. These sectors are responsible for a significant share of greenhouse gas (GHG) emissions and, at the same time, they could potentially play an important role in climate change mitigation. The agricultural sector alone is responsible for about 14% of total global anthropogenic GHGs emissions (UNFCCC, 2008) and is expected to have high emission growth rates, driven mainly by population and Sci. Agric. (Piracicaba, Braz.), v.67, n.1, p.31-40, January/February 2010 income increases. Agriculture provides the primary source of livelihood for more than one third of the world's total workforce (UNFCCC, 2008). In the heavily populated countries of Asia and the Pacific, up to half of the population works in the agriculture sector, while two thirds of the working populations in sub-Saharan Africa make their living from agriculture (FAO, 2009a;ILO, 2007). Agriculture is an important sector in terms of economy, employment and trade balance of the BRIC (Brazil, Russia, India and China) economies. The Food and Agriculture Organization of the United Nations (FAO) projects that agricultural production in developing countries will need to be increased by 70% in order to meet demands by 2050 (FAO, 2009b).

Deforestation generates an estimated additional 17% of emissions, thus resulting in a total contribution of the so called Land Use, Land-Use Change and Forestry (LULUCF) sector to nearly one third of the current total emissions. For major emerging countries with significant forest resources such Brazil, the LULUCF sector is the major source of emissions. In 2005 deforestation remains the top GHG contributor of Brazilian emissions with a share of nearly 52% [START_REF] Cerri | Brazilian greenhouse gas emissions: the importance of agriculture and livestock[END_REF]. [START_REF] Cerri | Brazilian greenhouse gas emissions: the importance of agriculture and livestock[END_REF] also calculated that GHG emissions for the agricultural sector increased by 38% from 1990 to 2005, and corresponded to 23% of 2005 total emissions, i.e. a total contribution of 75% for the LULUCF sector.

The LULUCF sector is now facing a set of inter-related challenges including providing livelihoods and food for growing populations, improving environmental sustainability, as well as the resilience and productivity of agro-ecosystems, meeting potential demand for bio-energy, including climate change adaptation and mitigation. The highly contentious negotiations surrounding a future LULUCF regime under the UNFCCC in a post-2012 international agreement could have significant impacts in enabling the sector to meet these challenges.

The mitigation potential of the LULUCF sector is high. Many of the technical options are readily available [START_REF] Bellassen | Multi-criteria spatialization of soil organic carbon sequestration potential from agricultural intensification in Senegal[END_REF][START_REF] Bernoux | Cropping systems, carbon sequestration and erosion in Brazil, a review[END_REF][START_REF] Cerri | Carbon cycling and sequestration opportunities in South America: The case of Brazil[END_REF][START_REF] Cerri | Tropical agriculture and global warming: impacts and mitigation options[END_REF][START_REF] Henry | Biodiversity, carbon stocks and sequestration potential in aboveground biomass in smallholder farming systems of western Kenya[END_REF] and could be deployed immediately: reducing emissions of carbon dioxide (CO 2 ) through the reduction of the rate of deforestation and forest degradation, adoption of improved cropland management practices (reduced tillage, integrated nutrient and water management); reducing emissions of methane and nitrous oxide through improved animal production, improved management of livestock waste, more efficient management of irrigation water on rice paddies, improved nutrient management; and, sequestering carbon (C) through conservation farming practices, improved forest management practices, afforestation and reforestation, agroforestry, improved grasslands management and restoration of degraded land. According to the Intergovernmental Panel on Climate Change (IPCC) the total technical mitigation potential of agriculture (considering all gases and sources) is estimated to be in the range 4.5 and 6 Gt CO 2 -eq per year in 2030 (IPCC, 2007;Smith et al. 2007a, b). Estimates indicate that many of these options are of relatively low cost, or generate significant co-benefits in the form of improved agricultural production systems, resilience and other ecosystem services [START_REF] Smith | Greenhouse gas mitigation in agriculture[END_REF].

As 74% of the agricultural mitigation potential lies in developing countries (Smith et al., 2007b), co-benefits related to rural poverty reduction and food security are of particular importance. Well designed forestry and agriculture development projects/programmes can play an important role in climate change mitigation, either by reducing emissions or by sequestering C. One of the main barriers to implement the potential of agricultural mitigation is the lack of methodologies or approaches that would help project designers and policy makers to integrate significant mitigation effects in agriculture and forestry development projects. The main objective of this study is to describe an easy-to-implement methodology proposed by the FAO to establish ex-ante greenhouse gas balances of agriculture and forestry development programs.

Description of EX-ACT

EX-ACT (EX-Ante C-balance Tool) is a tool developed by FAO to provide ex-ante measurements of the impact of agriculture and forestry development projects on GHG emissions and C sequestration, indicating its effects on the C balance. EX-ACT is a land-based accounting system, measuring C stocks, stock changes per unit of land, and CH 4 and N 2 O emissions expressed in t ha -1 of CO 2 -eq and t yr -1 of CO 2 -eq. The main output of the tool consists of the C-balance resulting from the difference between two scenarios: with and without project options. EX-ACT was designed to work at a project level but it can easily be up-scaled at programme/sector or national levels [START_REF] Cerri | Greenhouse gas mitigation options in Brazil for land-use change, livestock and agriculture[END_REF].

Minimum requirements towards quick integration in formulation and planning process -

The challenge was to provide a tool that is as simple as possible, as well as cost effective, but at the same time capable of covering the range of projects relevant for the LULUCF sector. To be recognized at the international level this tool must be compliant with international standards and best recognized practices. Moreover the tool should be readily understandable and usable by project developers. Finally, the tool must be upgradable over time, enabling the possibility of incorporating new information and policies, for instance, new approaches for Reducing Emissions from Deforestation and Forest Degradation (REDD) accounting that are currently being debated. Such requirements are preconditions to ensure quick scaling up towards the application of C-balance appraisal on new investment programmes. matic conditions of the project area together with basic data on land use, land use change and land management practices foreseen under projects activities as compared to a "business as usual" scenario. EX-ACT adopts a modular approach -each "Module" describing a specific land use -and following a three-step logical framework (Figure 1): (i) general description of the project (geographic area, climate and soil characteristics, duration of the project);(ii) identification of changes in land use and technologies foreseen by project components (deforestation, afforestation/reforestation, annual/perennial crops, rice cultivation, grasslands, livestock, inputs, energy); and (iii) computation of the C-balance with and without the project using IPCC default values andwhen available -ad-hoc coefficients.

Structure of EX-ACT -

Methodologies behind EX-ACT -

The first IPCC Guidelines for National Greenhouse Gas Inventories were published in 1995 and revised in 1996. They consist of three volumes (the reporting instructions, the workbook and the reference manual) each of which provides assistance to the analyst in the preparation of national GHG inventories [START_REF] Cerri | Brazilian greenhouse gas emissions: the importance of agriculture and livestock[END_REF]. Nevertheless some inconsistencies in the treatment of Agriculture and Land Use Change and Forestry (LUCF) were present. For instance, the chapter on agriculture covers only CH 4 and N 2 O for agricultural soils. Exchanges of CO 2 were incorporated in the LULUCF chapter even for agricultural soils [START_REF] Bernoux | CO 2 emission from mineral soils following land-cover change in Brazil[END_REF]. In 2003, the IPCC adopted a "Good Practice Guidance for Land use, Land-use Change and Forestry" (IPCC GPG for LULUCF), and UNFCCC decided that these guidelines should be used by Annex-I Parties (countries with mandatory emission reduction requirements) for preparing and reporting GHG inventories from the LULUCF sector. Non-Annex I Parties (e.g. Brazil) were only encouraged to apply them, as appropriate and to the possible extent, in the preparation of their GHG inventories in their national communications.

In the 2003 IPCC (2003) GPG for LULUCF, the land categories are divided into lands remaining in the same land use (for example, forest lands that remain as forest lands) and lands converted into another land-use category (for example, forest lands converted into croplands) during the inventory period. EX-ACT has adopted this approach.

Most of the steps proposed and the provided default coefficients are from the 2006 IPCC guidelines for National Greenhouse Gas Inventory (Thereafter referred as NGGI-IPCC-2006) and from Chapter 8 of the Fourth Assessment Report from working group III of IPCC (Smith et al., 2007a) for specific mitigation options not covered in NGGI-IPCC-2006. Other required coefficients are from published reviews or international databases. For instance embodied GHG emissions for farm operations, transportation of inputs, and irrigation systems implementation come from [START_REF] Lal | Carbon emission from farm operations[END_REF] and electricity emission factors are based on data from the International Energy Agency (IEA, www.iea.org). It should be highlighted that EX-ACT also allows users to incorporate their own more specific values when available.

Main concept -In a world where land availability is limited, competition amongst land uses is clearly an important aspect to be taken into account and dealt with in an adequate and comprehensive manner [START_REF] Riedacker | A global land use and biomass approach to reduce greenhouse gas emissions, fossil fuel use and to preserve biodiversity[END_REF]. Expanding the area of one crop or management practice in one locality is likely to have significant indirect impacts elsewhere. EX-ACT is based on the six broad categories (and sub-categories) proposed for reporting GHG inventories, but is focused mostly on three categories: Forest land, Cropland, Grassland.

Three approaches may be used to represent areas under a specific land use depending on the level of detail of the available information. The tool uses the approach (approach 2) of NGGI-IPCC-2006 that considers information on conversions between categories, but without full spatially-explicit location data. The final result of this approach can be represented as a land-use change matrix between categories.

When performing an ex-ante analysis the user should have an idea on what would happen without the project (i.e. the Business As Usual -BAU -Scenario or as referred to in this document as "Baseline"), thus the final balance is the comparison between the GHG emissions associated with the project compared with the baseline scenario.

The user can define two time periods, one for the implementation phase, i.e. the active phase of the project commonly corresponding to the funding and investment phase, and another for the capitalization phase, i.e. a period where the benefits of the investment are still occurring and may be attributed to the changes induced by the adoption of the project. In Figure 2, the user defines the duration of the implementation (i.e. t 1 -t 0 ) and capitalization phases (i.e. t 2 -t 1 ), the levels of key variables (hectares converted, stocking rates, amount of in-Figure 1 -Schematization of the modular EX-ACT structure.

IPCC defaults

Variables and coefficients

Related to land areas

Results (total, per ha, per ha and per year) puts…) for the current situation (x 0 ), at the end of the implementation phase for the baseline without project (x 1 ) or with the project (x 2 ). In terms of dynamics, changes associated with the rate of adoption of new practices or management options occur in the implementation phase. In contrast they are considered as no more change in the capitalization phase. The default assumption for the change is a linear function over time, but advanced users have the possibility to change the dynamic to either an "Immediate" or "exponential" function as shown in Figure 3.

The following example illustrates the impact of the choice of dynamic function on the final result: Given a situation (x 0 ) where farmers are using fertilizer on 100 ha of land, and the projected future use over the next 5 years (t 1 ) is forecasted to increase to 200 ha due to subsidies (x 1 ). Under the "immediate" option, the farmers will apply fertilizer on additional 100 ha from the first year of project implementation. In comparison, under the "Linear" dynamic, the growth in area under fertilizer increases progressively by 20 hectares per year. The exponential case represents an intermediary situation. The rate of change is faster at the beginning. The exponential approximation is defined by the equation Δ(t) = Δ max (1 -e -kt ), with Δ max = (x 1 -x 0 ), and k is set in order to have Δ(t 1 ) = 99% of Δ max . The immediate scenario corresponds to the maximum attainable change (100% level), while the linear dynamics corresponds to 50%, and the exponential an intermediary situation set to 78%.

Generic methodologies for estimating carbon pools changes (CO 2 balance) -Calculation of changes in C

pools is made using methods that can be applied in a very similar way for the type of land use change (i.e. generic methods). Generic methodologies are used mainly to account for changes between two categories during conversion, and concerns the five pools defined by IPCC guidelines and UNFCCC: above-ground biomass, below-ground biomass, soil, deadwood and litter. Most calculations, except where specified, use a Tier 1 approach with a stock-difference method for emission of CO 2 , calculated as the change of C stocks for the different pools, default values are proposed for each pool of each category (or subcategory or even main vegetation type):

• Above ground biomass: Default values correspond to estimates provided by NGGI-IPCC-2006 and expressed in t ha -1 of dry matter. The corresponding C stock (in t of C) is calculated using the specific C content, e.g. 0.47 for above-ground forest biomass.

• Below ground biomass: In most cases the below-ground biomass is estimated using a ratio (R) of below-ground biomass to above-ground biomass. EX-ACT uses the default values provided by NGGI-IPCC-2006, e.g. R is 0.37 for all tropical rainforest and 0.27 for Tropical mountain systems. In some cases the total above plus below ground biomass is used if it is not mandatory for calculation to have separate estimates.

• Litter and dead-wood: It is assumed that litter and dead wood pools are zero in all non-forest categories (excluding tree crops and perennial systems) and therefore transitions between non-forest categories involve no C stock changes in these two pools. For other transition default values are provided.

• Soil C: For the soil C estimates, the default values are based on default references for soil organic C stocks for mineral soils down to a depth of 30 cm. When Soil Organic C changes over time (land use change or management change), a default time period of 20 years is assumed for transitions between equilibria. These values are found in both IPCC 1997 and 2006 Guidelines, compiled from of a wide range of observations and data from long-term monitoring. Some modules use C change rates instead of the soil C stock difference and therefore do not require information on absolute soil C stock. In both approaches it is hypothesized that soil organic C stock changes during the transition to a new equilibrium SOC Three other sub-modules contain the list of variables, the default coefficients for all other modules, plus a specific module for GHG emission factors for electricity by country.

The description module contains a main description of the project boundaries. Users should define the main characteristics of various project components, i.e. dominant soil type, climate, duration of the project indicating both the length of implementation and capitalization phase and location of the project. The user is prompted for information that should either consist of a number or selection of a proposed list of choices. For instance for location, 11 options are available: Africa, Asia (Continental), Asia (Indian subcontinent), Asia (Insular), Middle East, Western Europe, Eastern Europe, Oceania, North America, Central America, and South America. Then the user should select which module to complete according to relevant activities of the project. Table 1 gives an overview of these modules. There are seven land modules and the information provided by the user in terms of surface concerned and variation are synthesized in the matrix module.

The deforestation module can be used to either quantify emissions from deforestation, or for the case of avoided deforestation. For this module, and the afforestation and reforestation module, different types of most probable vegetation are provided with their main characteristics according to the climatic information provided in the description module. For all land related modules, the user should input the main features of the associated land management, e.g. occurrence of burning. For Management Change (MC) Modules (see Table 1) mitigation strategies are proposed for each land management category. The main options available in EX-ACT are those widely accepted and documented in terms of C change rates (Smith et al., 2007a). These include practices concerning annual crops (e.g. nutrient management, tillage/residue management, irrigation, drainage that increases yields and/or generates higher inputs of C residue that can lead to increased soil C storage. It also covers perennial or semi-perennial systems, such as agroforestry and set-aside, and grazing land management. The "rice module" is related to flooded (permanently or part of the year) rice fields. This module covers CH 4 emission produced from anaerobic decomposition of organic matter and non-CO 2 GHG emissions (CH 4 and N 2 O) from biomass burning when occurring. In total, nine rice systems can be built with different water regimes and with corresponding emissions ranging from 0.24 to 2.47 kg ha -1 day -1 CH 4 . The grassland module deals with pasture degradation and rehabilitation with or without input management, and also incorporates options for fire occurrence.

Three modules are not specifically land based (in the sense there are not linked exclusively with a determined land area). One is concerned with livestock (and associated manure). An example of its use can be found in [START_REF] Cerri | Greenhouse gas mitigation options in Brazil for land-use change, livestock and agriculture[END_REF]. The second one (inputs) deals with emissions associated with fertilizer inputs and liming. It also includes GHG emissions from production, transportation, storage and transfer of agricultural chemicals using default values proposed by [START_REF] Lal | Carbon emission from farm operations[END_REF]. The last module (investment) is for performing GHG emissions associated with electricity consumption, with fuel consumption (agricultural or forestry machinery, genera-tors…), with installation of irrigation systems and with building of infrastructure. Detailed description of each module will be available in the associated handbook of EX-ACT and available early in 2010 on the FAO web site.

Cases studies

EX-ACT has been tested on several projects ranging from intensive crop and livestock production to sustainable agriculture, reforestation and rural development. In this section we present two case studies which are relevant to show the potential of EX-ACT to indicate projects effects on the C-balance: the "Accelerated Food Security Project" in Tanzania and the "Rio Rural Project" in Brazil.

An example from the "Accelerated Food Security

Project" in Tanzania -The case of the FAO/World Bank "Accelerated Food Security Project" (ASFP) in Tanzania represents an interesting example of the po-Sci. Agric. (Piracicaba, Braz.), v.67, n.1, p.31-40, January/February 2010 tential use of Ex-Act in estimating the impact of agricultural development projects on GHG emissions and C sequestration. The ASFP seeks to increase maize and rice production and productivity in targeted areas mainly by improving farmers access to critical agricultural inputs like fertilizers and improved seeds. Maize and rice production accounts respectively to 25 and 14% of agricultural GDP. Thus, improvements in food crop productivity will greatly contribute to the overall economic growth and poverty reduction in Tanzania.

Current productivity of maize and rice farmers in Tanzania is very low (e.g. in 2007/08 average maize yield was only 1.3 t ha -1 , much lower than in most neighbouring countries). The limited use of improved seeds and fertilizers has been the major reason of these low yields. In 2005/06, for example, the rate of fertilizer application in the country was reported at 8 kg ha -1 N while the depletion of soil nutrients was found to be about 61 kg ha -1 N (IFPRI, 2008). The use of improved seeds is also extremely low (only 24% of farmers, ranging from 15% in the South to 45% in the North), and the seeds planted are mainly self-produced and recycled (IFPRI, 2008).

Farmers participating in the project obtain, for an average of 0.5 hectare of maize/rice cropped area, an input package through the National Agricultural Input Voucher Scheme (NAIVS), consisting of one voucher for Nitrogen fertilizer (1 bag, i.e. 50 kg) Urea, one voucher for Phosphorus fertilizer (1 bag of diammonium phosphate -DAP -which is the most commonly used basal fertilizer in Tanzania) and one seed voucher (10 kg of open-pollinated varieties or hybrid maize and rice seeds). Participating farmers are also requested to abandon the practice of burning crop residues.

At full implementation the project is expected to benefit 2.5 million smallholders in several districts of Tanzania. The aim of NAIVS is to intensify food production in areas with high agro-ecological potential for pro- ducing staple crops in the southern and northern highlands as well as western regions: the southern highlands have the largest maize production zones in Tanzania (the four regions Mbeya, Rukwa, Ruvuma and Iringa account for more than 40%of the country's maize production), while rice production is more evenly distributed over the country. Project implementation will determine the adoption of improved cropland practices on most maize and rice cropland. It is expected that current land uses remain unchanged in the "without project" scenario, while "with project" around 70% of targeted area will adopt improved cropland management practices for both crops. The following table describes the current land use and the two alternative scenarios ("with" and "without project").

o s l a t u b , N 2 * O x i r t a M I G G N f o ) 5 r e t p a h C ( 4 e m u l o V ) 6 0 0 2 , C C P I ( d n a l s s a r GC M g n i n i a m e r d n a l s s a r G d n a l s s a r G O C y l t s o M 2 o s l a t u b , H C 4 N d n a * 2 * O x i r t a M I G G N f o ) 6 r e t p a h C ( 4 e m u l o V ) 6 0 0 2 , C C P I ( k c o t s e v i L e s u d n a l a t o N y r o g e t a c H C 4 N d n a 2 O I G G N f o ) 0 1 r e t p a h C ( 4 e m u l o V e m u l o V d n a ) 6 0 0 2 , C C P I ( e h t f
( I G G N f o 1 e m u l o V f o t n e m t
Conventional maize production makes use of farmsaved and recycled seeds without any nutrient management practices and the practice of incorporating crop residues is generally not adopted, while, there are two main types of rice production systems in the project area: rainfed lowland, often flooded (with non flooded preseason longer than 180 days); and irrigated, continuously flooded, with a flooded preseason longer than 30 days. Improved cropland management practices will consist of incorporating crop residues into soils, using improved seed varieties and implementing a nutrient management plan. Specifically, the following rates of fertilization will be adopted both on maize and rice fields: 0.1 t ha -1 yr -1 of Urea; 0.018 t ha -1 yr -1 of N synthetic fertilizer and 0.046 t ha -1 yr -1 of phosphorus synthetic fertilizer. Given the cropland area estimated in the "with project" scenario, overall rates of fertilizer with project correspond at the end of the implementation phase to 83,820 t yr -1 of urea, 15,087 t yr -1 of N (as DAP), and 38,557 t yr -1 of P (as DAP).

The impact of project activities on the overall C balance has been estimated using the modules called Annual, Rice and Inputs of the EX-ACT tool. The latter computed CO 2 emissions from urea applications together with the CO 2 -eq emissions from production, transportation, storage and transfer of agricultural chemicals (phosphorus and nitrogen synthetic fertilizers, i.e. DAP). The increase in soil organic content due to the adoption of improved crop management practices is taken into account in the module Annual, while the specific module on rice estimated the effects on the C balance of improved rice management practices. Results show that annual crops with project compared to without project result in a net sink of 11.25 Mt CO 2 -eq, although expanded fertilizer use and changes in rice management are net sources of respectively 1.6 Mt CO 2 -eq and 2.4 Mt CO 2 -eq. The adoption of improved land and integrated nutrient management practices will contribute to soil C sequestration so that the net project effect will be the creation of a C sink, with positive effects in terms of mitigation. The overall C balance of the ASFP of Tanzania is computed as a difference between C sinks and sources and it has been estimated at 7.2 Mt CO 2 -eq over 20 years, corresponding to 350,000 t yr -1 of CO 2eq Given the area of 1.06 Mha, this corresponds to a sequestration rate of 0.3 t ha -1 yr -1 of CO 2 -eq. As expected, the most significant source of mitigation comes from the implementation of improved cropland management in maize production.

The case of the "Rio Rural Project" in Brazil -Ex-Act can be applied to more complex projects such as the GEF/World Bank "Rio Rural Project" in Brazil, aimed at promoting sustainable rural development in North and Central Rio de Janeiro State to cope with the ongoing land erosion and its adverse effects on rural economy. The project supports a broad set of activities and its implementation is based on an integrative approach of the agro-silvo-pastoralist systems, based on hydrographical micro-catchments as planning units. This section describes the application of EX-ACT to some project components identified as having a significant impact on the C balance and for which data were available. The case also represents an interesting example of the potential of using the tool during the project design phase.

EX-ACT has been used to estimate the GHG impact of the following components of the project: (i) Protection of springs and streams; (ii) Protection of the legal reserves; (iii) Promotion of sustainable cropping patterns: reduced tillage, contour/strip cropping, inter-cropping, mulching, improve water management, use of green manure and organic fertilizer; (iv) Pasture management: pasture rotations and production of sugarcane The results of the EX-ACT simulations are reported here, showing the GHG impact of each module and the overall effect of the project on the C balance (Figure 4).

All the land-based activities supported by the project contribute to create a C sink totalling 564,399 t CO 2 -eq after 20 years, the only source arising from construction of infrastructure (9,930 t CO 2 -eq after 20 years). Overall, the net effect on the C balance of the project components taken into account amounts to over 550,000 t CO 2eq. However, this result needs to be interpreted with caution since other project components which are likely to impact negatively on the C balance were not taken into account at this stage due to a lack of data. For example, the project is expected to increase the use of lime and fertilizers and to promote road construction. Also, extension activities (400 technicians are expected to work extensively in the area) needed to implement the project and promote the adoption of sustainable land management practices will increase GHG emissions because of the increase in fuel consumption. The case is also interesting to show that, if used at an early stage, EX-ACT provides an interesting planning tool to mainstream climate change mitigation considerations into agricultural and rural development projects, allowing project designers to simulate different scenarios and to analyze the relative impact on C balance. For example, project designers for the Rio Rural Project are currently testing the impact on C balance of three different scenarios of fuel consumption, depending on the type of fuel used: 100% gasoline, 20% ethanol in gasoline, and 100% ethanol technologies.

The way forward

Agricultural systems can contribute significantly to an overall mitigation that will help to reduce the extent of adaptation required and catastrophic impacts on systems and sectors, on which lives and livelihoods depend. Many agricultural mitigation options, particularly those that involve soil C sequestration also generate co-benefits for adaptation, food security and rural development. These agricultural mitigation options can be pursued in the context of, and without adverse affects to, national sustainable development processes. Mitigation benefits from agriculture could generate an additional financing stream to support such developments through the implementation of payment for environmental service programs (PES), or potentially through international agreements on mitigation financing for developing countries currently under negotiation in the UNFCCC.

Until now there has been no user-friendly tool to allow for a quick appraisal of the potential mitigation impacts of agricultural investment projects. EX-ACT allows for the C-balance appraisal of new investment programmes by ensuring an appropriate method available to donors and planning officers, project designers, and decision makers within agriculture and forestry sectors in developing countries. The tool can also help to identify the mitigation impacts of various investment project options and thus provide an additional criterion for consideration in project selection.

EX-ACT has gone through a field-testing program in 2009 and a peer-review process is ongoing. The Beta version will be release and available on the web early 2010 for use by project designers in international organizations and donor agencies working on agriculture (and forestry) development and/or involved in agriculture (and forestry) investment projects. The tool will then be available for free use by donors and technical partners. However appropriate training, software updating and technical quality monitoring framework will have to be in place to support effective implementation. After an initial phase of use at the project and program level, the tool will also be modified and tested for use in the development of national sector strategies and policies (e.g. to compute the C balance of aggregated agriculture sector strategies and policy options) or for regional initiatives.

  EX-ACT consists of a set of 18 linked Microsoft Excel sheets into which project designers insert information on dominant soil types and cli-Sci. Agric. (Piracicaba, Braz.), v.67, n.1,p.31-40, January/February 2010

Figure 2 -

 2 Figure 2 -Schematic representation of how the final balance is calculated with EX-ACT.

  Land_Use Change, 2 MC = Management Change, *from biomass burning (Chapter 2 of NGGI -IPCC, 2006), 3 NGGI = Guidelines for National Greenhouse Gas Inventories. Sci. Agric. (Piracicaba, Braz.), v.67, n.1,p.31-40, January/February 2010

Figure 4 -

 4 Figure 4 -Facsimile of the results as exhibited in EX-ACT.

  Although soil C changes in response to management changes may often be best described by a non linear function, the linear assumption greatly simplifies the methodology and provides a good approximation over a multi-year period. O and CH 4 are either associated with a specific land use category or subcategory (e.g. CH 4 emissions from rice), or are estimated at project aggregated data (e.g. emissions from livestock and N 2 O emission from fertilizers). CH 4 and N 2 O emissions are converted into CO 2 -eq emissions based on the global warming potential of each gas. The user has the ability to use either the official values under the Kyoto Protocol of the UNFCCC, or the last update provided by the IPCC (2007).

		Generic methodologies for non-CO 2 GHG -For N 2 O and CH 4 emissions, the generic approach consists of multi-plying an emission factor for a specific gas or source cat-
		egory with activity data related to the emission source
		(e.g. area, animal numbers or mass unit…). Emissions
		of N 2
	Variable considered	(ha of land use, number of cattle heads,…)	x 0 x 1 x 2	Implementation phase	Capitalization phase	With project Without project Difference use to compute the final C balance in tons CO 2 equivalent (reduced emissions + C sequestered)
							Time (years)
			t 0			t 1	t 2
				Variable considered	(x)	"Immediate"	"Linear"	"Exponential"	Time
					t 0		t 1	t 0	t 1	t 0	t 1	(years)

Figure 3 -Schematic representation of the dynamics implemented in EX-ACT. Sci. Agric. (Piracicaba, Braz.), v.67, n.1,p.31-40, January/February 2010 occurs with a linear pattern.

Short description of the (sub)modules -

Three sub-modules on dominant soil type, climatic characteristics and soil ecological zone allow for more detailed accounting of the project situation in response to individual user needs. The climate module provides external internet links to additional resources useful for determining local or regional climate with greater precision. These resources include a software to download and useful for estimating climate using a database built on 28800 weather stations (LocClim, available at http:// www.fao.org/nr/climpag/pub/en3_051002_en.asp).

Table 1 -

 1 Overview of Modules used for C-Balance calculation.

	o e l e m u d o a M N	f	h t	e		c	n i r y o a M g e t a					y r Id e o g e t a c -C C P n r e c n o c G H G	k d n i e t L a l r e h t o e r	h t i w u d o M -l e d n a L	a M	n i	e m	h t	o	o d	o l	e i g	s	a	n	d	e r	e f	e r	n	c	e	s
	D	e	o f	e r	s	a t	o i t	n			L	U	C	1	o t : d n d e s a l e t r u -d e v n a n l o t d n o r c e a l t h d t n o a n L a s e r o F	y l t H C 4 s o M	O d , 2 n a C *	s l * a O t u N b 2	o	e l x i u d r t a o M M d C M n a	s	e m 2 , C u l C o P V I (	C ) 6 ( 0 4 0	h	a	p	e t	r	4	)	o	f	N	G	G	I
	n n o o i t i t a a t t s s e e r r o o f f A d n a f e R			L	U	C	e t r u -d e v n a n l d n a l t s o r c e e r h d t n o a n L a o F o t	d e s	o t a l :	n	d	y l t H C 4 s o M	O d , 2 n a C *	s l * a O t u N b 2	o	e l x i u d r t a o M M d C M n a	s	e m 2 , C u l C o P V I (	C ) 6 ( 0 4 0	h	a	p	e t	r	4	)	o	f	N	G	G	I
																							L	a	n	d		c	o	n	e v	e t r	d	o t
	h t a h O C	L e g r n e	a	n	d	U	s	e	L	U	C	s d u -d n a l n t s a l e r r o e F h t -n o o n n a -n o n r e h t o n a	: o t e	O C 2	e l x i u d r t a o M M d C M n a	s	e m 2 , C u l C o P V I (	C ) 6 ( 0 4 0	h	a	p	e t	s r	4	6 -	)	o	f	N	G	G	I
																							o F	e r	s	a l t	n	d
	s AC M 2 p o r C l a u n n		C C	o r o r	p p	a l a l	n n	d d	e r	a m	n i	n i	g	y l t H C 4 s o M	O d 2 , n a C *	s l * a O t u N b 2	o	a M	x i r t	o e h V t e h t	r t e t r o p p a e h R C ) a 7 0 0 2 , ( t . n " e n m o i t s s a g i t i M e s s A h " t r e m u o u l f l a t e h t i m S ( C C P I	) f 8 o	o	f
	s l e r a i o f n n o r g A e r e P	/ p o y r t r C	s	C M		C C	o r o r	p p	a l a l	n n	d d	e r	a m	n i	n i	g	y l t H C 4 s o M	O d 2 , n a C *	s l * a O t u N b 2	o	a M	x i r t	o e h V t e h t	r t e t r o p p a e h R C ) a 7 0 0 2 , ( t . n " e n m o i t s s a g i t i M e s s A h " t r e m u o u l f l a t e h t i m S ( C C P I	) f 8 o	o	f
	e RC M c i		C C	o r o r	p p	a l a l	n n	d d	e r	a m	n i	n i	g	o M	s	y l t	C	H	4

Table 2 -

 2 Land use systems corresponding to the current situation and the two alternative scenarios ("with" and "without project") for the FAO/World Bank "Accelerated Food Security Project" (ASFP) in Tanzania.

	e Ln o s u d n a i t a u t i s t n e r r u Ct c e j o r p t u o h t i Wt c e j o r p h t i W
																																-----
	a M	e z i	c	o	n	e v	n	o i t	n	a	l	e t	c	h	n	o	o l	y g	9	7	2	6 .	9	7	2	6 .	2	6	0	1 .
	y g M0 o l o n h c e t d e v o r p m i e z i a	0	7	1	2	5 .
	c i R	e	a r	n i	e f	d	c	o	n	e v	n	o i t	n	a	l	e t	c	h	n	o	o l	y g	6	2	9 .	6	2	9 .	1	8	9 .
	y g R9 . o l o n h c e t l a n o i t n e v n o c d e t a g i r r i e c i 2 29 . 2 29 . 6
	c i R	e	a r	n i	e f	d	p m i	o r	e v	d	e t	c	h	n	o	o l	y g	0	0	4	4	0 .
	y g R0 o l o n h c e t d e v o r p m i d e t a g i r r i e c i	0	1	6	1 .

. Agric. (Piracicaba, Braz.), v.67, n.1, p.31-40, January/February 2010 forage

  ; (v) Increased use of lime and fertilizers; (vi) Promotion of small agro-industry: construction of warehouse and packing house, installation of cooling systems and; (vii) Improvement of infrastructure: construction of sanitations and roads. Table3presents how each component is taken into account in the EX-ACT simulation when data were available.

Table 3 -

 3 Description of the components and Module concerned in EX-ACT with information regarding the systems considered and their extent for the GEF/World Bank "Rio Rural Project" in Brazil.

																															e r A	a	c	o	n	c	e	n r	e	d
	t Pd e n e n o p m o C t c e j o r n r e c n o c ) s ( e l u d o m T C A -X Et i n U	C	u	e r r	n	t	W	h t i	o	u	t	W	h t i
																															s	u t i	a	o i t	n	o r P	e j	c	t	o r P	e j	c	t
	o i t s m c e t a o i t c e t e v r e o r P e r t s o r P s e r	n n	o o	f f		n l a a g g e n i r l e p h s t	d	n / n o i t o i t a t a t s s e r e r o f o f e f D Aa h0 a h < n o i t a t s e r o f e R d n a l s s a r G a h 0 1 5 8	8	0 0 5	0	1 0 < 5 8 0
	o r P	o m	o i t	n	o	f	s	u	s	a t	n i	a	b	e l	: As m s p o r c l a u n n e t s y s t n e r r u Ca h0 7 0 , 6 2 24 0 1 , 5 2 24 9 9 , 0 2 2
	c	o r	p	p	n i	g		p	a	e t t	n r	s						p m I	o r	e v	d	s	y	s	e t	s m	h	a	0	9	6	6	5	0 ,	7	6
																															: d Gd e n a l s s a r d a r g e d y l e t a r e d o Ma h1 9 65 8 50
	a P	s	u t	e r	a m	n	a	e g	e m	n	t			d e m e v e g o r a n p m I a m	/ w t n	n i	p	u	s t	N	o	n	e d	a r g	e d	d	a a h0 h 0	5 1 8 21 0 0 9 2 4
	-f o o r g t n a e l l m a m e v s n o i t a t i n a s s o r f p o m n I o i t y r o m t s u o r P d n i ( e r u t c u r t s a r f n i	)	n I f o	t u n r t e m t s a r f s n e v i	C u t ( c	s ) s n e o r	u r t	c	o i t	n	m 2	0	1	3	9 ,	0	0	3	1	7 ,	2	0

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EX-ACT (Ex-Ante Carbon-balance Tool, ferramenta para o balanço ex-ante de Carbono) sua maioria provenientes do 4º Relatório de Avaliação do IPCC ( 2007