ON THE ASYMPTOTIC BEHAVIOR OF DENSITY OF SETS DEFINED BY SUM-OF-DIGITS FUNCTION IN BASE 2

Transcription

1 #A58 INTEGERS 7 (7) ON THE ASYMPTOTIC BEHAVIOR OF DENSITY OF SETS DEFINED BY SUM-OF-DIGITS FUNCTION IN BASE Jordan Emme Aix-Marseille Université, CNRS, Centrale Marseille, Marseille, France jordanemme@univ-amufr Alexander Prikhod ko Moscow Institute of Physics and Technology, Moscow, Russia sashaprihodko@gmailcom Received: 4//5, Revised: 7/4/6, Accepted: /5/7, Published: //7 Abstract Let s (x) denote the number of occurrences of the digit in the binary expansion of x in N We study the mean distribution µ a of the quantity s (x + a) s (x) for a fixed positive integer a It is shown that solutions of the equation s (x + a) s (x) = d are uniquely identified by a finite set of prefixes in {, }, and that the probability distribution of di erences d is given by an infinite product of matrices whose coefficients are operators of l (Z) Then, denoting by l(a) the number of occurrences of the pattern in the binary expansion of a, we give the asymptotic behavior of this probability distribution as l(a) goes to infinity, as well as estimates of the variance of the probability measure µ a Introduction Background In this article we are interested in the statistical behavior of the di erence of the number of digits in the binary expansion of an integer x before and after its summation with a This kind of question can be linked with carry propagation problems developed in [5] and [9], and is linked to computer arithmetic as in [8] or [3], but our approach is di erent The main definitions of the paper are the following

2 INTEGERS: 7 (7) Definition For every integer x in N whose binary expansion is given by we define the quantity x = nx x k k, x k {, }, k= s (x) = nx x k, which is the number of occurrences of the digit in the binary expansion of x Definition For every integer x in N whose binary expansion is given by: x = k= nx x k k, k= we denote by x the word x x n in {, } Remark 3 Since we are working in the free monoid {, } of binary words, let us state right away that we will denote the cylinder set of a word w = w w n by the standard notation [w] := {v {, } N v v n = w w n } These sets form a topological basis of clopen sets of {, } N for the product topology Moreover, we endow the set of binary configurations {, } N with the natural probability measure P that is the balanced Bernoulli probability measure defined on the Borel sets The function s modulo was extensively studied for its links with the Thue- Morse sequence (as found in [4]), for example, or for arithmetic reasons as in [] and [6] In this paper our motivation is not of the same nature Thus we will not look at s modulo, but instead at the function s, which is the sum of the digits in base In [], Bésineau studied the statistical independence of sets defined via functions such as s, ie sum of digits functions To this end he studied the correlation function defined in the following way Definition 4 Let f : N! C Its correlation f : N! C is the function, if it exists, defined by NX f (a) = lim f(k)f(k + a) N! N k= This correlation was studied in [] for functions of the form k 7! e i s(k) where s is a sum-of-digits function in a given base In the case of base, this motivates us to understand the following equation, with parameters a in N and d in Z: s (x + a) s (x) = d ()

3 INTEGERS: 7 (7) 3 Namely, we wish to understand, for given integers a and d, the behavior of the density of the set {x N s (x + a) s (x) = d}, which always exists (from [, 7] for instance) We remark that we can define in this way, for every integer a, a probability measure µ a on Z defined, for every d, by µ a (d) := lim N! N # {x apple N s (x + a) s (x) = d} Example 5 Let us compute µ and deduce µ n for every n First, we remark that s (x + ) s (x) apple for every non-negative integer x, since there is only one in the binary expansion of It is easy to see that µ () is the density of the set of even integers, so we have µ () = We also remark that in order to get a di erence d apple, one has to propagate a carry in the summation d + times In other words, suppose we wish to have s (x + ) s (x) = d with a negative d; then x has to start with d+ This means that x d+ mod ( d+ ) The set {x N x d+ mod ( d+ )} is of density d+ Finally, if d > µ (d) = if d apple d+ Now notice that µ a = µ a for every a This is proved later with Proposition 8 Hence, for every n in N, if d > µ n(d) = if d apple d+ Section studies the set {x N s (x + a) s (x) = d} with a combinatorial approach, using constructions from theoretical computer science such as languages, graphs, and automata In this way we prove that the probability µ a is given by a product of matrices Section 3 is a technical part that allows us to understand the asymptotic behavior of such a measure as a gets bigger in a certain, non-trivial, sense This section develops the idea that our probability measure gets smaller as l(a) increases Namely, we give an upper bound of the l (Z) norm of the probability measure µ a depending on the number of occurrences of in a Finally, we give bounds on the variance of µ a, also depending on the number of patterns in a Results The main results are the following Theorem The distribution µ a is calculated via an infinite product of matrices whose coe cients are operators of l (Z) applied to a vector whose coe cients are

4 INTEGERS: 7 (7) 4 elements of l (Z): µ a = (Id, Id) A an A an A a A a, where the sequence (a n ) nn is the binary expansion of a, is the Dirac mass in, the A i are defined by Id A = S S, A = S S Id, and S is the left shift transformation on l (Z) Remark We recall that the set of finite measures on Z is in bijection with the elements of l (Z) We will always identify finite measures on Z and elements of l (Z) Such a result allows an analytical study of these distributions as the binary expansion of a becomes more and more complicated Let us formalize this notion of complexity for a Definition For every a in N, let us denote by l(a) the number of subwords in the binary expansion of a Remark 3 We advise the reader to be careful, as this notion of complexity has nothing to do whatsoever with the classical one in the field of language theory We are thus interested in what happens as there are more and more patterns in the word a We can estimate precisely the asymptotic behavior of the l (Z) norm k k of this distribution as a goes to infinity by increasing the number of subwords Namely, we have the following theorem Theorem There exists a real constant C such that for every integer a we have the following: kµ a k apple C l(a) /4 Finally, in the last section, we wish to obtain a much more precise result regarding the behavior of such a distribution than just estimates of the l norm So we study the way in which the variance of the random variable of probability law µ a is linked to the number of subwords that are in the binary expansion of a We define the variance of µ a to be the quantity Var(a) = X dz µ a (d)(n m a, ), where m a, is the first moment (the mean) of the probability measure µ a It is shown in this last section that these moments (the mean and the variance) do exist; in fact, the mean is always zero Moreover, we have bounds on this variance as shown in the following result

5 INTEGERS: 7 (7) 5 Theorem For every integer a such that l(a) is large enough, the variance of µ a, denoted by Var(a), has bounds l(a) apple Var(a) apple (l(a) + ) Statistics of Binary Sequences In this section we wish to understand the following quantity: µ a (d) := lim N! N # {x apple N s (x + a) s (x) = d}, for every given positive integer a and every integer d We know from [] that such a limit exists, and it has been studied in [7], for example, but here we give a proof using the structure of solutions of the following equation: s (x + a) s (x) = d for every a and d Let us investigate such solutions, as well as their construction, in order to understand the distribution of probability µ a of di erences d = s (x + a) s (x) We prove that this distribution is given by an infinite product of matrices whose sequence is given by the binary expansion of a Combinatorial Description of Summation Tree First we prove the following lemma Lemma For all a in N and d in Z, there exists a finite set of words P a,d = {p,, p k } {, } such that x is solution of () if and only if x k[ [p j ] j= where, for every word w in {, }, [w] denotes the cylinder set of words whose prefix is w Remark Before we prove Lemma, let us remark that for every even integer n the following holds: n s (n) = s

6 INTEGERS: 7 (7) 6 We also remark that, for every odd integer n, the following holds: n s (n) = s + Proof Let us prove this lemma by induction on a It is obviously true that for a =, there exists a (possibly empty) set of words for every d in Z describing the solutions of Equation () Let us assume that it is true for every integer not greater than some given a in N Now let k be an integer not greater than a If k is even then, for every d, P k,d = {w, w P k,d} [ {w, w P k,d} Indeed, x being an even solution of s (x + k) s (x) = d is equivalent to having x + k x s s = d which can be written as s x + k x s = d From the induction hypothesis it follows that x starts with a followed by a word in P k,d Moreover, x being an odd solution of s (x + k) s (x) = d is equivalent to having s x which can be written as + k + s x + k x s + = d x s = d Then, by the induction hypothesis, x must start with a followed by a word in P k,d If, however, k is odd, then P k,d = {w, w P k,d } [ {w, w P k+,d+} Indeed, x being an even solution of s (x + k) s (x) = d is equivalent to having which can be written as x + k x s + s = d s x + k x s = d

7 INTEGERS: 7 (7) 7 From the induction hypothesis it follows that x starts with a followed by a word in P k,d Moreover, x being an odd solution of s (x + k) s (x) = d is equivalent to having x + k + x s s + = d which can be written as s x + k + x s = d + Then, by the induction hypothesis, we can state that x must start with a followed by a word in P k+,d+ Hence knowing the sets P,d for every integer d allows to inductively compute the sets P a,d for every a in N Remark 3 It should be noted that such a set is not unique For instance, the sets {, } and {} can both qualify as P, with the lemma s notations From the previous lemma it follows immediately that, for all integers d and positive integers a, the sequence (# {x apple N s (x + a) s (x) = d}) NN can be written as a sum of sequences, all of which contain an arithmetic progression Thus the following limit exists: µ a (d) := lim N! N # {x apple N s (x + a) s (x) = d} Moreover, a quick computation yields µ a (d) = [ pp a,d [p] where P is the balanced Bernoulli probability measure on {, } N Remark 4 In all that follows, we always consider sets P a,d such that for every words p, p in P a,d, p 6= p =) [p ] \ [p ] = ; () So in this case, µ a (d) = X A pp a,d P([p]) Remark 5 Note that a su cient condition for a set P a,d to satisfy () is that all the prefixes be of same length since two cylinders defined by prefixes of the same length are disjoint whenever the prefixes are di erent

8 INTEGERS: 7 (7) 8 The proof of Lemma naturally gives the idea of an inductive way to compute the prefixes A comfortable way to do that is to span, for every a, a tree a in such a way that the family of trees obtained is consistent with the following recursive properties for every k: P k,d = {w, w P k,d } [ {w, w P k,d } and P k+,d = {w, w P k,d } [ {w, w P k+,d+ } Lemma 6 For each a in N, there exists a tree a with vertices labeled in {,, a} Z and edges labeled in {, } such that, for every d in Z, words in P a,d are exactly paths labels from the vertex (a, ) to a vertex (, d) Proof Let us prove this by induction on a First, assume that a = The tree is given by Figure Vertices labels are a pair, the first coordinates are boxed (in the vertex), the second coordinates of the labels are circled The edges labels ( or ) are indicated above the edges One can check that the path from the vertex labeled (, ) (which is the root in this case) to a vertex (, d) defines a sequence of edges whose labels form a word which is the only word in P,d For example, if s (x + ) s (x) =, then x indeed begins with the word Figure : Infinite subtree spanned by every vertex labeled by Let us now assume that every such tree exists until a fixed integer a Let k be an integer no greater than a

9 INTEGERS: 7 (7) 9 If k is even, we add to the tree k/ a vertex labeled by (k, ) and add two edges between the vertices (k, ) and ( k, ): one labeled by a and the other one by a This tree does satisfy the property that the set P k,d is the set of paths labels from (k, ) to a vertex (, d) Indeed, P k,d = {w, w P k/,d } [ {w, w P k/,d }, and starting from vertex (k, ), one can choose either edge ( or ) to get to the vertex ( k, ), from which the induction hypothesis holds If k is odd, we define g k to be a copy of the tree k where every vertex labeled (n, c) is relabeled (n, c + ) We also denote by g k+ the copy of k+ where every vertex is relabeled from (n, c) to (n, c ) The tree k is obtained by taking the union of g k and g k+, adding a vertex labeled (k, ) and two edges: one labeled between (k, ) and ( k, ) and one labeled between (k, ) and ( k+, ) We recall that P k,d = {w, w P k,d } [ {w, w P k+,d+} Hence using the induction hypothesis on both k+ for k and k yields the result Thus for every integer k there exists a tree k which satisfies the property described in the lemma If one wishes to construct such a tree for a given positive integer a, let us give a method to easily do so (which is a direct consequence of the proof) Let us choose a particular a in N and construct the associated tree a The construction starts at level with only one vertex labeled by (a, ) and we define the tree level by level If a is defined up to level n in N, from every vertex we construct the vertices at level n + in the following way Starting from a vertex labeled by (k, c): if k is even we add a vertex at level n + labeled by ( k, c) and add one edge of each type between these vertices; if, however, k is odd, we add two vertices labeled by ( k and we construct an edge of type between (k, c) and ( k edge of type between (k, c) and ( k+, c ) k+, c+) and (, c ), c + ) and an Notice that the second coordinate of the labels of the vertices can be seen as a counter that keeps track of the di erence s (x + a) s (x) as the summation is processed digit by digit More precisely, if we actually compute by hand the sum of x and a, the value of the counter on the n th level is exactly the number of s

10 INTEGERS: 7 (7) lost or gained when the n th digit of the sum is computed Once we reach a vertex (, d), the summation is completed We can see an example of such a construction on Figure for a = 5 Figure : Part of the tree 5 Example 7 Let a = 5 Then we start our construction on the first level of the tree by creating the root labeled by (5, ) The number 5 being odd, we add the vertices (,) and (3,-) on the second level, and add an edge of type between vertices (,) and (5,) and an edge of type between vertices (3,-) and (5,) We continue the construction on the third level by: adding a child to vertex (,) labeled by (,), since is even; adding two children to vertex (3,-) labeled by vertices (,) and (,-), since 3 is odd The edge between vertices (,) and (3,-) is of type and the edge between vertices (,-) and (3,-) is of type We continue this process on each vertex at each level to obtain the tree of Figure Let us compute the words in P 5, It can be read from the tree 5 that this set is given by the words,, and We wish to remark that, because of the way x is defined, the word, for example, is not the binary expansion of 6 but of, as everything is mirrored We now remark that having such a family of trees proves the following proposition Proposition 8 For every a in N and every d in Z, we have the following identities: µ a (d) = µ a (d)

11 INTEGERS: 7 (7) and µ a+ (d) = µ a(d ) + µ a+(d + ) Proof For every positive integer a and every integer d, we can read from the tree a that words in P a,d are exactly words beginning with either or, then followed by a word in P a,d Thus we have µ a (d) = µ a (d) Notice that words in P a+,d are exactly the words beginning with a followed by a word in P a,d and the words beginning with a followed by a word in P a+,d+ It follows that µ a+ (d) = µ a(d ) + µ a+(d + ) Remark 9 This proposition allows us to explicitly compute every distribution µ a, since it is trivial to give explicit closed formulas for µ and µ It also gives an understanding of the link between µ a and the binary expansion of a However, we prefer another presentation of such a result Namely, one that assembles in a close formula these identities as we follow the binary decomposition of a Such a formula is given in Theorem One way to do such a thing is to collapse the tree a in order to obtain a collapsed graph This is more or less done by identifying the vertices whose label s first coordinates are the same, and we obtain a graph whose structure is studied in the next section Let us define properly this collapsed graph for an integer a The graph has levels indexed on N For every n in N, the level n has exactly two vertices labeled by b a c and n b a c + n There are edges only between vertices on consecutive levels and they are described in the following way: if b a c is even then there are two edges between b a n c and b a n c There n+ is also an edge between b a c+ and b a n c, and another between b a n+ c+ n and b a c + ; n+ if b a n c is odd then there are two edges between b a n c + and b a There is also an edge between b a n c and b a b a n c and b a n+ c + ; c + n+ c, and another between n+

12 INTEGERS: 7 (7) Figure 3: Collapsing 5 Figure 4: Tail of a collapsed graph Remark If one wants to see this graph as actually being a collapsing of the tree by identifying vertices on the same level which have the same labels, one should be careful as there are two ambiguities First of all, for coherence purposes which will appear latter, we add a vertex a + on level which does not appear on the tree Secondly, notice that for n big enough, a collapsed graph on levels greater than n is given in Figure 4 Such a tail of a collapsed graph is, a priori, di erent from what one would get by identifying vertices on levels greater than n on the tree Indeed, there should not be edges between vertices labeled by on the collapsed graph We deliberately chose to do it this way in order to control the length of the prefixes in P a,d The reason for such a choice appears in the next subsection Looking at this graph after collapsing is quite interesting since it gives us an understanding of the link between the summation process and the binary expansion of a It also allows the statistical study of the behavior of s (x + a) s (x) An

13 INTEGERS: 7 (7) 3 example of the tree 5 collapsing is given in Figure 3 Distribution of s (x + a) s (x) The goal of this section is to prove the following theorem Theorem The distribution µ a is calculated via an infinite product of matrices whose coe cients are operators of l (Z): µ a = (Id, Id) A an A an A a A a, where the sequence (a n ) nn is the binary expansion of a, is the Dirac mass in, the A i are defined by Id A = S S, A = S S Id, and S is the left shift transformation on l (Z) This theorem follows from observations on the collapsed graph First, notice that only two di erent patterns can appear between any two levels of the collapsed graph We add labels on the edges of the collapsed graph in a way consistent with the tree a Those patterns with the labels on edges are given by Figure 5 Notice also that the order in which the patterns appear is given by the binary expansion of a Now, we remark that we can identify words in {, } with paths in the collapsed graph whose starting point is the vertex labeled by a on level Definition We define the functions E : {, }! {, } and C : {, }! Z in the following way: E(w) = if the path associated to w has endpoint the vertex of smaller label of its level (the leftmost vertex on Figure 5); E(w) = if the path associated to w has endpoint the vertex of greater label of its level (the rightmost vertex on Figure 5); C(w) is the counter of the path associated to w (in the same way as for the tree defined in the previous subsection) The counter of the empty word is and the + and on the edges of the graphs in Figure 5 indicate whether to increment or decrement the counter In other words, C(w) is the number of edges of type + that the path associated to w contains minus the number of edges of type Notice that the way the counter is defined is consistent with the counter on the tree a for every integer a

14 INTEGERS: 7 (7) 4 Definition 3 Let us define, for each positive integer a, each level n of the collapsed graph the probability measures a,n and a,n on every d in Z: a,n(d) = n #{w {, }n E(w) = and C(w) = d}, a,n(d) = n #{w {, }n E(w) = and C(w) = d} This seems to only define a sequence in l (Z), but we recall that we can identify finite measures on Z with sequences in l (Z) It is a quick check to verify that we actually have defined probability measures Notice also that, for every d in Z, a,n(d) is the asymptotic density of the set of integers whose binary expansion starts with a word in {w {, } n E(w) = and C(w) = d} Figure 5: The two di erent patterns encountered in the collapsed graph Lemma 4 The i a,n are given by induction: a,n+ a,n+ = A a,n an a,n where a = +X k= a k k Proof We wish to compute a,n+ and a,n+ It is obvious that the pattern we see between levels n and n + is given by a n If a n = then we encounter the left pattern of Figure 5 In this case we have This being true for all d, we can write a,n+(d) = a,n(d) + a,n(d ) a,n+ = a,n + S ( a,n),

15 INTEGERS: 7 (7) 5 where S is the left shift transformation on the space l (Z) For the same pattern we also have a,n+(d) = a,n(d + ) which can be rewritten as a,n+ = S( a,n) So in this particular case we can write the relations in the following way: a,n+ Id a,n+ = S a,n S a,n The same arguments for the pattern on the right of figure 5 (which corresponds to the case where a n = ) yields a,n+ a,n+ = S a,n S Id a,n Lemma 4 allows us to prove Theorem : Proof We recall that for every a in N and every d in Z, a set P a,d is given by paths in the tree a whose start is the root labeled by a and whose end is a vertex labeled by (, d) Let P a,d = {p,, p j } be this set of words, with p j being the prefix of maximal length Since we defined the collapsed graph in a way that is consistent with the tree a (namely, the edges have the same labels), for every i in [, j ], the word p i represents a path starting from a and ending on a vertex labeled by in the collapsed graph and satisfying E(p i ) = and C(p i ) = d We denote by m d the length of p j, and consider the set of prefixes {p,, p k } {, } m d such that j[ k[ [p i ] = [p k] i= Notice that every element of {p,, p k } has a prefix that is an element of {p,, p j } Hence, for every i in [, k], we have E(p i ) = and C(p i ) = d and thus: i= {p,, p k} {w {, } m d E(w) = and C(w) = d} Let us prove the reciprocal inclusion Let w be a word in {, } m d such that E(w) = and C(w) = d Since E(w) =, and since w is of length m d, its ending vertex is labeled by a (on the collapsed graph) Let us denote by p the minimal prefix of w such that the ending vertex of p is a Then necessarily, C(p) = d Now

16 INTEGERS: 7 (7) 6 notice that such a path p is also a path in a with ending vertex labeled by (, d) Then p is in {p,, p j } So w is in {p,, p k } Consequently, {p,, p k} = {w {, } m d E(w) = and C(w) = d} and thus, by Remark 5, a,m d (d) = µ a (d) Finally, we have the following: µ a (d) = (Id, Id)A amd A amd A a A a, a a, (d) It is also worth noting that, after rank m d, continuing to multiply on the left by A does not change the final result, so this allows for a more convenient expression: µ a (d) = This proves the theorem (Id, Id) A an A an A a A a, a a, (d) Example 5 Let us show that Theorem allows us to find the expression of µ that was computed in the introduction According to the theorem, µ = (Id, Id) A A A Now A = whence and it can easily be seen that, for every integer k, A k = µ a = P k j= j+ ( )k k X j= j+ j Now, for a more involved example, let us assume we wish to compute µ 3 () Well, µ 3 = (Id, Id) A A A A, and then a quick computation yields A A = j!

17 INTEGERS: 7 (7) 7 and so A A A = Now in order to know µ 3 (), we have to multiply by A once more: A A A A = Note that multiplying by A again will never change the value of the coe cient of in the first coordinate, and neither will appear in the second coordinate (since we apply S to the second coordinate when multiplying by A ) Hence µ 3 () = 5 6 In the general case, for given a in N and d in Z, we should first compute A an A a, which gives a vector whose coordinates are linear combinations of Dirac measures Then we should multiply by A until the Dirac measures on the second coordinate have mass only on integers no greater than d, which ensures that the coe cient of d in the first coordinate is exactly µ a (d) 3 Asymptotic Properties of Distributions µ a In this section we study the asymptotic behavior of the family of distributions µ a as l(a) increases We prove that the l (Z) norm of µ a tends to as l(a) increases, and hence the densities of µ a tend to zero Let us recall that l(a) denotes the number of distinct patterns in the binary expansion of a Remark 36 For every integer a there exists k integers p,, p k such that a = p p p k or a = p p p k depending on the parity of a If a is even, k = l(a), and else, k = l(a) + In other words, the number of distinct continuous maximal blocks of digits in a is either l(a) + or l(a) 3 Convergence to Zero We start by proving the following theorem Theorem 3 There exists a constant C such that for every integer a we have the following: kµ a k apple C l(a) /4 The proof is established in a series of lemmas Let us recall the notation Id A = S S, A = S S Id,

18 INTEGERS: 7 (7) 8 where S is the left shift on l (Z) Let us define a norm on the space of n n matrices Y = (y i,j ) i,j{,,n} by the following: ky k = max applejapplen i= nx y i,j Remark 3 We remark right away that this defines a submultiplicative norm We consider the Fourier transform ˆµ a defined for every in [, ) by ˆµ a ( ) = X dz e id µ a (d) Notice that, given an element f of l (Z), for every in [, ), Hence, since µ a (d) = (Id, Id) A an A an A a A a (d) S(f)( ) ˆ i = e ˆf( ) we get that ˆµ a ( ) = (, ) Âa n ( )Âa n ( ) Âa ( )Âa ( ) where  ( ) := e i,  ( ) := ei e i ei Lemma 33 We have the following elementary identities for every : kâ( )k = kâ( )k = kâ( )Â( )k = kâ( )Â( )k = kâ( )Â( )Â( )k = kâ( )Â( )Â( )k := Notice that is strictly less than except when = Proof For every in [, ), we have and So we have  ( )Â( )Â( ) = e i + 4 ( ) = +p 5+4 cos 4 4 e i + 8 e i + 4, 4 ei 8 ei + 4 ei  ( )Â( )Â( ) = 8 e i + 4 e i 4 ei + 8 ei e i ei + 4 kâ( )Â( )Â( )k = kâ( )Â( )Â( )k

19 INTEGERS: 7 (7) 9 Moreover, the triangular inequality yields 4 e i + 8 e i ei + 8 ei apple 4 e i + 8 e i ei + 8 ei, so we have 4 e i + 8 e i ei + 8 ei apple 4 ei + e i + 4 Hence the sum of the moduli of the terms in the first column is less than the sum of the moduli of the terms of the second column, which implies that kâ( )Â( )Â( )k = 4 e i + ei + 4 so finally kâ( )Â( )Â( )k = + p cos 4 Lemma 34 For every k in N and every in [, ) kâ( ) (Â( )) k  ( )k apple kâ( ) Â( ) Â( )k Proof First let us state that, for all positive integer k, for all in [, ), e ik (Â( )) k = k k( ) where k( ) = e i We can compute that, for every k in nn,  ( ) (Â( )) k  ( ) = e ik + e i k kx j= e ij j (3) k( ) e i(k+) k+ + e i e i k( ) 4 k ( ) + ei 4 4 k( ) + 4C A Moreover, noticing that for every positive integer k and every in [, ), implies that e i(k+) k+ + e i k+( ) = e i = e i(k+) k+ + e i 4 k( ) + ei k+( ) + ei k+( ) k( ) k ( ) + ei 4

20 INTEGERS: 7 (7) This means that the sum of moduli of coe cient on the first column of matrix  ( ) (Â( )) k+  ( ) is equal to the sum of the moduli of the coe cients of the second column of the matrix Â( ) (Â( )) k  ( ) Hence, to prove that, for every integer k and all in [, ), kâ( ) (Â( )) k  ( )k apple kâ( ) Â( ) Â( )k it is enough to actually show the following: e ik k + e i k( ) + ei k( ) apple kâ( ) Â( ) Â( )k Using the triangle inequality, we have e ik k + e i k( ) + ei k( ) apple k + k( ) which, still using the triangle inequality, again gives e ik k + e i k( ) + ei k( ) apple k + ei + kx 4 + j j=3 and So we get e i + e i e ik k + e i which completes the proof since ( ) + ei k( ) + ei ( ) apple + ei + 4 k( ) apple ei e i = kâ( ) Â( ) Â( )k Lemma 35 If apple apple then Hence, for every positive integer N, with defined as in Lemma 33 ( ) apple e /5 k N k apple ke N /5 k = /4 5 N

21 INTEGERS: 7 (7) Proof The proof is left to the reader Proof of Theorem 3 We remind the reader that µ a = µ a, + µ a, is the sum of the components of the vector µ a, which is calculated as an infinite product µa, µ a = = A µ an A an A a A a a, Let us represent the binary expansion of a as a sequence of l(a) groups of separated by zeros: a = m m m l(a) We apply Lemma 34 to half of the patterns and then use the bound of Lemma 33: kâ( ) (Â( )) k  ( )k apple ( ) We can do this only on half of s because we need to avoid problematic patterns like, since we need at least two between two blocks on to be able to apply Lemma 34 twice but we have only one in between Thus we get k Âa n ( ) Âa ( )k apple ( ) N, N = l(a), and hence, for each in [, ) and j in {, },! ˆµ a,j ( ) apple ( ) N = e N /5, and kµ a,j k = p kˆµ a,j k apple p k N k apple p s Z e Nt /5 dt = by Lemma 35 Now, N = l(a) yields kµ a k apple p 5 (l(a) ) /4 = O l(a) /4 /4 5 8 N Remark 36 It follows directly from Theorem 3 that the density µ a (j)! as l(a)! Let us also remark that this theorem actually gives important information concerning the fact that the probability measure µ a is linked with the complexity of the binary expansion of a (which is measured by l(a)) In fact, for every integer n, µ n = µ, so it is possible to find arbitrarily large a such that µ a does not tend to zero (actually whenever l(a) does not tend to infinity)

22 INTEGERS: 7 (7) 3 Asymptotic Mean and Variance of µ a Let us first recall the following:  ( ) := e i,  ( ) := ei e i, ei for every in [, ) We begin by using Taylor s expansion on the matrices Â( ) and Â( ) to get  k ( ) = I k + k + k + O( 3 ), k {, }, where I =! I =!, = i, = i and we observe that the following relations hold!, = 4!, = 4!,!, (, )I k = (, ), (4) (, ) k = (, ), (, ) = (, ), (, ) = (, ) (5) Lemma 3 For all in [, ), the product of matrices (Â( )) k converges as k goes to + We denote the limit  ( ), and further we have the following equality:  ( ) =! i e e i Proof We recall that k is defined in Section 3, Equation (3) The lemma is proved by remarking the following:! (Â( )) k = k( ) e ik k Now the asymptotic expansion of  is where  ( ) = I + + O( 3 ),! I =, =!

23 INTEGERS: 7 (7) 3 Definition 3 Given a in N of binary length N, we define the following product: a ( ) =  Âa N ( ) Âa ( ) Let us recall that, for every a in N, Lemma 33 Let v = asymptotic expansion: where and, for all j in {,, N + }, ˆµ a ( ) = (, ) a ( ) The product (, ) a ( ) v has the following (, ) a ( ) v = + V (a) + O( 3 ), V (a) = (, ) ( v N+ + an v N + + a v ), v j = I aj I a v In other words, this Lemma 33 states that µ a has mean and variance V (a) Indeed, the moments of a probability measure are given by the successive derivatives near of its Fourier transform Proof Let us first recall that Moreover, (, )I = (, )I = (, )I = (, ) (, ) a ( ) v = (, ) I + + O( 3 ) where the product is defined as: NY i= I ai + ai + ai + O( 3 ) v, NY M i := M N M i= Hence the constant term in the asymptotic expansion of (, ) a ( ) v is (since (, ) = ) In addition, we have (, ) = (, ) = (, ),

24 INTEGERS: 7 (7) 4 so the term of degree is With the same argument we can compute the quadratic term V (a) Notice that every coe cient or is killed by left multiplication by (, ) Hence the terms of the form I I aj+ aj I aj I ai+ ai I ai I a disappear after left multiplication by (, ), and thus do not contribute in any way to the coe cient of the quadratic term So the only terms contributing to V (a) in the product Â( )Âa N ( ) Âa ( ) are the terms with coe cient a j with j in [, N ] and the term with coe cient Definition 34 Let a be a positive integer with binary expansion a = a a n For every j [, n], let b j := max {k + a j k = = a j } and b j := b j b j In other words, b j is the length of a consecutive and maximal sequence of digits equal to a j to the left of j (including position j) in a and b j is the length of the next block of identical digits to the left of the block containing the digit a j For instance, if a =, then b 6 = 4 since a a 6 = and b 6 = Lemma 35 For every j in {,, N}, the value (, ) aj v j is estimated as follows apple (, ) aj v j apple, b j b j Proof Assume that a j = (the proof for the case a j = is similar) Observe that I and I contract the segment [(, ), (, )] to the first second point respectively with a contraction factor of More precisely, x x I x [, ] = x x x b j apple, and x x I x [, ] = x x x Hence, with t = j b j b j, I t I [(, ), (, )] apple,

25 INTEGERS: 7 (7) 5 Moreover, the block I b j maps [(, ), (, )] to the segment [( b j, ( b j )), (, )] Then the block I b j contracts the latter to h i b j + ( b j ), b j + ( b j ), b j +, b j + Multiplying on the left by (, ) aj = (, ) = (, ) means taking the second coordinate with an additional coe cient /, and the lemma then follows Remark 36 The same proof yields (, ) v N+ apple b N Moreover, we have the following lemma Lemma 37 NX l(a) apple j= b j apple l(a) + Proof This lemma is obtained by noticing that each block of identical digits in the binary expansion of a spans a term P N j= which is a partial sum of a geometric b j sequence of ratio and first term, so each is smaller than and greater than In fact, let us assume that there are k blocks in the binary expansion of a of lengths p,, p k (ie, a = p p p k p k, for example) Then NX j= b j = kx Xp i i= j= j and kx i= kx apple Xp i i= j= j apple kx Moreover, k can be equal to either l(a) + or l(a), which completes the proof i= Theorem 38 For every integer a, the variance V (a) of µ a has bounds l(a) apple V (a) apple (l(a) + ) Proof The idea of this theorem is to see that each block of same digits in the binary expansion of a has an impact on V (a) estimated by a constant Let us estimate the value of V (a) By Lemma 35 we have that V (a) = NX NX (, ) aj v j (, ) v N+ apple j= j= b j + b N

26 INTEGERS: 7 (7) 6 Moreover, the upper bound of Lemma 37 yields V (a) apple (l(a) + ) Now we will prove the lower bound: Note that V (a) (l(a) ) V (a) = NX X N (, ) aj v j (, ) v N+ (, ) aj v j j= j= so, using Lemma 35, we have: V (a) NX j= b j b j Whence, V (a) NX j= b j NX j= b j +b j Let us denote by k the largest integer such that a = = a k Noticing that b j = for all j in {,, k}, we have that V (a) NX j=k+ b j NX j=k+ b j +b j Moreover, for every integer j larger than k, b j, hence V (a) NX j=k+ b j from which we derive V (a) (l(a) ) by applying the lower bound of Lemma 37 This proves Theorem 38 On a final note, it is interesting to remark that the bounds in Theorem 38 might not be optimal but that for a sequence of integers (a(n)) nn such that V (a(n)) lim n! l(a(n)) = + and such that the limit of l(a(n)) exists, then this limit can take di erent values in the interval [, 4] We give some examples of computer simulations in Figure 6

27 INTEGERS: 7 (7) 7 V (a(n)) l(a(n)) for a(n) = nx k= k V (a(n)) for a(n) = l(a(n)) nx 4k + 4k+ k= V (a(n)) l(a(n)) for a(n) = nx k= 6k + 6k+ 6k+ V (a(n)) + for a(n) = n l(a(n)) Figure 6: Asymptotics of the ratio V (a(n)) l(a(n)) for di erent sequences (a(n)) nn V (a(n)) It would be interesting to understand the asymptotic behavior of l(a(n)) Having a necessary condition for convergence and a precise idea of how it behaves asymptotically would help in understanding the variance of the probability measure µ a Acknowledgements We wish to thank the referee whose very helpful remarks played an important role in the improvement and clarification of this article We also wish to thank Nicolas Bedaride and Pascal Hubert whose advice and careful readings of this article helped greatly

28 INTEGERS: 7 (7) 8 References [] J Bésineau, Indépendance statistique d ensembles liés à la fonction somme des chi res, Séminaire Delange-Pisot-Poitou, 3e année (97/7), Théorie des nombres, Fasc, Exp No 3, page 8, Secrétariat Mathematique, Paris, 973 [] M Drmota, C Mauduit, and J Rivat, Primes with an average sum of digits, Compos Math 45() (9), 7 9 [3] M Ercegovac and T Lang, Digital Arithmetic, Elsevier, 3 [4] M Keane Generalized Morse sequences, Z Wahrscheinlichkeitstheorie und Verw Gebiete (968), [5] D Knuth, The average time for carry propagation, Indag Math (Proceedings) 8() (978), 38 4 [6] C Mauduit and J Rivat, Sur un problème de Gelfond: la somme des chi res des nombres premiers, Ann of Math (), 7(3) (), [7] J Morgenbesser and L Spiegelhofer, A reverse order property of correlation measures of the sum-of-digits function, Integers (), A47 [8] J Muller, Arithmétique des Ordinateurs, Masson, 989 [9] N Pippenger, Analysis of carry propagation in addition: An elementary approach, Journal of Algorithms 4() (),