On distribution functions of ξ(3/2) n mod 1

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1 ACTA ARITHMETICA LXXXI. (997) On distribution functions of ξ(3/2) n mod by Oto Strauch (Bratislava). Preliminary remarks. The question about distribution of (3/2) n mod is most difficult. We present a selection of known conjectures: (i) (3/2) n mod is uniformly distributed in [, ]. (ii) (3/2) n mod is dense in [, ]. (iii) (T. Vijayaraghavan []) lim sup{(3/2) n } lim inf n n {(3/2)n } > /2, where {x} is the fractional part of x. (iv) (K. Mahler [6]) There exists no ξ R + such that {ξ(3/2) n } < /2 for n =,, 2,... (v) (G. Choquet [2]) There exists no ξ R + such that the closure of {{ξ(3/2) n }; n =,, 2,...} is nowhere dense in [, ]. Few positive results are known. For instance, L. Flatto, J. C. Lagarias and A. D. Pollington [3] showed that lim sup{ξ(3/2) n } lim inf n n {ξ(3/2)n } /3 for every ξ >. G. Choquet [2] gave infinitely many ξ R for which /9 {ξ(3/2) n } /9 for n =,, 2,... R. Tijdeman [9] showed that for every pair of integers k and m with k 2 and m there exists ξ [m, m + ) such that {ξ((2k + )/2) n } for n =,, 2,... 2k The connection between (3/2) n mod and Waring s problem (cf. M. Bennett []), and between Mahler s conjecture (iv) and the 3x + problem (cf. [3]) is also well known. 99 Mathematics Subject Classification: K3. This research was supported by the Slovak Academy of Sciences Grant 227. [25]

2 26 O. Strauch In this paper we study the set of all distribution functions of sequences ξ(3/2) n mod, ξ R. It is motivated by the fact that some conjectures involving a distribution function g(x) of ξ(3/2) n mod may be formulated as in (i) (iv). For example, the following conjecture implies Mahler s conjecture: If g(x) = constant for all x I, where I is a subinterval of [, ], then the length I < /2. The study of the set of distribution functions of a sequence, still unsatisfactory today, was initiated by J. G. van der Corput []. The one-element set corresponding to the notion of asymptotic distribution function of a sequence mod was introduced by I. J. Schoenberg [8]. Many papers have been devoted to the study of the asymptotic distribution function for exponentially increasing sequences. H. Helson and J.-P. Kahane [4] established the existence of uncountably many ξ such that the sequence ξθ n does not have an asymptotic distribution function mod, where θ is some fixed real number >. I. I. Piatetski-Shapiro [7] characterizes the asymptotic distribution function for the sequence ξq n mod, where q > is an integer. For a survey, see the monograph by L. Kuipers and H. Niederreiter [5]. In Section 2, we recall the definition of a distribution function g and we define a mapping g g ϕ associated with a given measurable function ϕ : [, ] [, ]. The formula defining g g ϕ was used implicitly by K. F. Gauss for ϕ(x) = /x mod in his well-known problem of the metric theory of continued fractions (g ϕ is given e.g. in [5, Th. 7.6]). The induced transformation between derivatives g g ϕ is the so-called Frobenius Perron operator. In Section 3, choosing ϕ(x) as f(x) = 2x mod and h(x) = 3x mod, we derive a functional equation of the type g f = g h, for any distribution function g of ξ(3/2) n mod. As a consequence we give some sets of uniqueness for g, where X [, ] is said to be a set of uniqueness if whenever g = g 2 on X, then g = g 2 on [, ], for any two distribution functions g, g 2 of ξ(3/2) n mod (different values of ξ R, for g, g 2, are also admissible). From this fact we derive an example of a distribution function that is not a distribution function of ξ(3/2) n mod for any ξ R. We also conjecture that every measurable set X [, ] with measure X 2/3 is a set of uniqueness. An integral criterion for g to satisfy g f = g h is also given. In Section 4, we describe absolutely continuous solutions g of functional equations of the form g f = g and g h = g 2 for given absolutely continuous distribution functions g, g 2. In Section 5, we summarize the examples demonstrating all the above mentioned results. 2. Definitions and basic facts. For the purposes of this paper a distribution function g(x) will be a real-valued, non-decreasing function of the

3 Distribution functions of ξ(3/2) n mod 27 real variable x, defined on the unit interval [, ], for which g() = and g() =. Let x n mod, n =, 2,..., be a given sequence. According to the terminology introduced in [5], for a positive integer N and a subinterval I of [, ], let the counting function A(I; N; x n ) be defined as the number of terms x n, n N, for which x n I. A distribution function g is called a distribution function of a sequence x n mod, n =, 2,..., if there exists an increasing sequence of positive integers N, N 2,... such that A([, x); N k ; x n ) lim = g(x) for every x [, ]. k N k If each term x n mod is repeated only finitely many times, then the semiclosed interval [, x) can be replaced by the closed interval [, x]. Every sequence has a non-empty set of distribution functions (cf. [5, Th. 7.]). A sequence x n mod having a singleton set {g(x)} satisfies A([, x); N; x n ) lim = g(x) for every x [, ] N N and in this case g(x) is called the asymptotic distribution function of a given sequence. Let ϕ : [, ] [, ] be a function such that, for all x [, ], ϕ ([, x)) can be expressed as the union of finitely many pairwise disjoint subintervals I i (x) of [, ] with endpoints α i (x) β i (x). For any distribution function g(x) we put g ϕ (x) = g(β i (x)) g(α i (x)). i The mapping g g ϕ is the main tool of the paper. A basic property is expressed by the following statement: Proposition. Let x n mod be a sequence having g(x) as a distribution function associated with the sequence of indices N, N 2,... Suppose that each term x n mod is repeated only finitely many times. Then the sequence ϕ({x n }) has the distribution function g ϕ (x) for the same N, N 2,..., and vice versa every distribution function of ϕ({x n }) has this form. P r o o f. The form of g ϕ (x) is a consequence of A([, x); N k ; ϕ({x n })) = i A(I i (x); N k ; x n ) and A(I i (x); N k ; x n ) = A([, β i (x)); N k ; x n ) A([, α i (x)); N k ; x n ) + o(n k ). On the other hand, suppose that g(x) is a distribution function of ϕ({x n }) associated with N, N 2,... The Helly selection principle guarantees a suit-

4 28 O. Strauch able subsequence N n, N n2,... for which some g(x) is a distribution function of x n mod. Thus g(x) = g ϕ (x). It should be noted that if all of the intervals I i (x) are of the form [α i (x), β i (x)), then o(n k ) = and the assumption of finiteness of repetition is superfluous. In this paper we take for ϕ(x) the functions f(x) = 2x mod and h(x) = 3x mod. In this case, for every x [, ], we have g h (x) = g( with inverse functions and g f (x) = g(f (x)) + g(h 2 f (x) = x/3, h 2 (x)) + g(f2 (x)) g(/2), (x)) + g(h 3 (x) = x/2, f2 (x) = (x + )/2, (x)) g(/3) g(2/3), (x) = (x + )/3, h 3 (x) = (x + 2)/3. 3. Properties of distribution functions of ξ(3/2) n mod. Piatetski- Shapiro [7], by means of ergodic theory, proved that a necessary and sufficient condition that the sequence ξq n mod with integer q > has a distribution function g(x) is that g ϕ (x) = g(x) for all x [, ], where ϕ(x) = qx mod. For ξ(3/2) n mod we only prove the following similar property. Theorem. Every distribution function g(x) of ξ(3/2) n mod satisfies g f (x) = g h (x) for all x [, ]. P r o o f. Using {q{x}} = {qx} for any integer q, we have {2{ξ(3/2) n }} = {3{ξ(3/2) n }}. Therefore f({ξ(3/2) n }) and h({ξ(3/2) n }) form the same sequence and the rest follows from the Proposition. The above theorem yields the following sets of uniqueness for distribution functions of ξ(3/2) n mod. Theorem 2. Let g, g 2 be any two distribution functions satisfying (g i ) f (x) = (g i ) h (x) for i =, 2 and x [, ]. Set I = [, /3], I 2 = [/3, 2/3], I 3 = [2/3, ]. If g (x) = g 2 (x) for x I i I j, i j 3, then g (x) = g 2 (x) for all x [, ]. P r o o f. Assume that a distribution function g satisfies g f = g h on [, ] and let J i, J j, J k be the intervals from [, ] described in Figure.

5 Distribution functions of ξ(3/2) n mod 29 x J J 2 J 3 J 4 /3 2/3 J 2 J 4 J 6 J 3 J /2 J J 3 J 5 J 2 y (x) f (x) h 2 (x) f 2 (x) 3 (x) /5 3/5 /3 2/3 Fig. There are three cases of I i I j.. Consider first the case I 2 I 3. Using the values of g on I 2 I 3, and the equation g f = g h on J, we can compute g( (x)) for x J. Mapping x J to x J 2 by using (x) = f (x ), we find g(f (x)) for x J 2. Then, by the equation g f = g h on J 2 we can compute g( (x)) for x J 2; hence we have g(f (x)) for x J 3, etc. Thus we have g(x) for x I. 2. Similarly for the case I I In the case I I 3, first we compute g(/2) by using g f (/2) = g h (/2), and then we divide the infinite process of computation of g(x) for x I 2 into two parts: In the first part, using g(y), for y I I 3, and g f = g h on [, ], we compute g( 2 (x)) for x J. Mapping x J x J 2 by 2 (x) = f (x ) and employing g f = g h we find g( 2 (x)) for x J 2. In the same way this leads to g(f 2 ) on J 3, g( 2 ) on J 3, g(f ) on J 4, g( 2 ) on J 4, and so on.

6 3 O. Strauch Similarly, in the second part, from g on I I 3 and g f = g h on [, ] we find g( ) on J, g(f2 ) on J 2, g( 2 ) on J 2, g(f ) on J 3, g( 2 ) on J 3, etc. In both parts these infinite processes do not cover the values g(2/5) and g(3/5). The rest follows from the equations g f (/5) = g h (/5) and g f (4/5) = g h (4/5). Next we derive an integral formula for testing g f = g h. Define F (x, y) = {2x} {3y} + {2y} {3x} {2x} {2y} {3x} {3y}. Theorem 3. A continuous distribution function g satisfies g f = g h on [, ] if and only if \ \ F (x, y) dg(x) dg(y) =. P r o o f. Let x n, n =, 2,..., be an auxiliary sequence in [, ] such that all (x m, x n ) are points of continuity of F (x, y), and let c X (x) be the characteristic function of a set X. Applying c [,x) (x n ) = c (xn,](x), we can compute \ ( N c N f ([,x))(x n ) N 2 c N ([,x))(x n )) dx where n= n= F f,h (x, y) = max(f(x), h(y)) + max(f(y), h(x)) max(f(x), f(y)) max(h(x), h(y)) = N 2 N m,n= F f,h (x m, x n ), = 2 ( f(x) h(y) + f(y) h(x) f(x) f(y) h(x) h(y) ). Applying the well-known Helly lemma we have \ \ \ (g f (x) g h (x)) 2 dx = F f,h (x, y) dg(x) dg(y) for any continuous distribution function g. Here 2F f,h (x, y) = F (x, y). 4. Inverse mapping to g (g f, g h ) Theorem 4. Let g, g 2 be two absolutely continuous distribution functions satisfying (g ) h (x) = (g 2 ) f (x) for x [, ]. Then an absolutely continuous distribution function g(x) satisfies g f (x) = g (x) and g h (x) = g 2 (x)

7 Distribution functions of ξ(3/2) n mod 3 for x [, ] if and only if g(x) has the form g(x) = Ψ(x) for x [, /6], Ψ(/6) + Φ(x /6) for x [/6, 2/6], Ψ(/6) + Φ(/6) + g (/3) Ψ(x 2/6) + Φ(x 2/6) g (2x /3) + g 2 (3x ) for x [2/6, 3/6], 2Φ(/6) + g (/3) g (2/3) + g 2 (/2) Ψ(x 3/6) + g (2x ) for x [3/6, 4/6], Ψ(/6) + 2Φ(/6) + g (/3) g (2/3) + g 2 (/2) Φ(x 4/6) + g (2x ) for x [4/6, 5/6], Ψ(/6) + Φ(/6) + g (/3) + Ψ(x 5/6) Φ(x 5/6) g (2x 5/3) + g 2 (3x 2) for x [5/6, ], where Ψ(x) = T x ψ(t) dt, Φ(x) = T x φ(t) dt for x [, /6], and ψ(t), φ(t) are Lebesgue integrable functions on [, /6] satisfying ψ(t) 2g (2t), φ(t) 2g (2t + /3), 2g (2t) 3g 2(3t + /2) ψ(t) φ(t) 2g (2t + /3) + 3g 2(3t), for almost all t [, /6]. P r o o f. We shall use a method which is applicable for any two commuting f, h having finitely many inverse functions. The starting point is the set of new variables x i (t): x (t) := f h f(t) = f f h(t), x 2 (t) := f 2 h f(t) = f2 f h(t), x 3 (t) := f 3 h f(t) = 2 f2 f h(t), x 4 (t) := f2 h f(t) = 2 f f h(t), x 5 (t) := f2 2 h f(t) = 3 f f h(t), x 6 (t) := f2 3 h f(t) = 3 f2 f h(t). Here the different expressions of x i (t) follow from the fact that f(h(x)) = h(f(x)), x [, ]. For t [, /6] we have x i (t) = t + (i )/6, i =,..., 6. Substituting x = j h f(t), j =, 2, 3, into g f (x) = g (x), and x = f i f h(t), i =, 2, into g h (x) = g 2 (x) we have five linear equations for g(x k (t)), k =,..., 6. Abbreviating the composition f i j h f(t) as f h 2 hf(t), and x i(t) as x i, we can write g(x ) + g(x 4 ) g(/2) = g ( hf(t)), g(x 2 ) + g(x 5 ) g(/2) = g ( 2 hf(t)), g(x 3 ) + g(x 6 ) g(/2) = g ( 3 hf(t)), g(x ) + g(x 3 ) + g(x 5 ) g(/3) g(2/3) = g 2 (f fh(t)), g(x 2 ) + g(x 4 ) + g(x 6 ) g(/3) g(2/3) = g 2 (f 2 fh(t)).

8 32 O. Strauch Summing up the first three equations and, respectively, the next two equations, we find the necessary condition g (/3) + g (2/3) + 3g(/2) + (g ) h (hf(t)) for t [, /6], which is equivalent to and = (g 2 ) f (fh(t)) + g 2 (/2) + 2(g(/3) + g(2/3)) g (/3) + g (2/3) g 2 (/2) = 2(g(/3) + g(2/3)) 3g(/2) (g ) h (x) = (g 2 ) f (x) for x [, ]. Eliminating the fourth equation which depends on the others we can compute g(x 3 ),..., g(x 6 ) by using g(x ), g(x 2 ), g(/3), g(/2), and g(2/3) as follows: () g(x 3 ) = g(/3) + g(2/3) g(/2) g(x ) + g(x 2 ) g ( 2 hf(t)) + g 2(f fh(t)), g(x 4 ) = g(/2) g(x ) + g ( hf(t)), g(x 5 ) = g(/2) g(x 2 ) + g ( 2 hf(t)), g(x 6 ) = g(/3) + g(2/3) g(/2) + g(x ) g(x 2 ) g ( hf(t)) + g 2(f 2 fh(t)), for all t [, /6]. Putting t = and t = /6, we find g(/2) = 2g(/3) 2g(/6) + g (/3) g (2/3) + g 2 (/2), g(2/3) = 2g(/3) 3g(/6) + 2g (/3) g (2/3) + g 2 (/2). These values satisfy the necessary condition g (/3) + g (2/3) g 2 (/2) = 2(g(/3) + g(2/3)) 3g(/2). Moreover, g(/3) = g(x 2 (/6)), g(/6) = g(x 2 ()), and thus g(x 3 ),..., g(x 6 ) can be expressed by only using g(x ), g(x 2 ). Next, we simplify () by using hf(t) = ff 2 h 2 f 2 f 2 hf(t) = ff 2 h fh(t) = hh 2 f hf(t) = ff h fh(t) = hh 3 f hf(t) = f(x 4) for g(x 4 ), 2 hf(t) = f(x 5) for g(x 5 ), fh(t) = h(x 3) and 2 hf(t) = f(x 2) for g(x 3 ), 2 fh(t) = h(x 6) and hf(t) = ff h hf(t) = f(x ) for g(x 6 ). Now, each g(x i ) can be expressed as g(x), x [(i )/6, i/6]. To do this we use the identity x i (x j (t)) = x i (t) for t [, ] and i, j 6,

9 Distribution functions of ξ(3/2) n mod 33 which immediately follows from the fact that For example, f i j hff k h l hf(t) = f i j hf(t). g(x 3 ) = g(/3) + g(2/3) g(/2) g(x ) + g(x 2 ) g (f(x 2 )) + g 2 (h(x 3 )), for t [, /6], which is the same as g(x) = g(/3) + g(2/3) g(/2) g(x (x)) + g(x 2 (x)) g (f(x 2 (x))) + g 2 (h(x)) for x [2/6, 3/6]. In our case x (x) = x i/6 and x 2 (x) = x + /6 i/6 for x [i/6, (i + )/6] and i =,..., 5. Now, assuming the absolute continuity of g(x ) and g(x 2 ) we can write t\ g(x (t)) = ψ(u) du, g(x 2 (t)) = /6 \ t\ ψ(u) du + φ(u) du for t [, /6]. Summing up the above we find the expression g(x) in the theorem. For the monotonicity of g(x) we can investigate g (x i (t)) for t [, /6] and i =,..., 6, which immediately leads to the inequalities for ψ and φ given in our theorem. 5. Examples and concluding remarks. Define a one-jump distribution function c α : [, ] [, ] such that c α () =, c α () =, and { if x [, α), c α (x) = if x (α, ]. The distribution functions c (x), c (x), and x satisfy g f (x) = g h (x) for every x [, ]. 2. Taking g (x) = g 2 (x) = x, further solutions of g f = g h follow from Theorem 4. In this case ψ(t) 2, φ(t) 2, ψ(t) φ(t), for all t [, /6]. Putting ψ(t) = φ(t) =, the resulting distribution

10 34 O. Strauch function is for x [, 2/6], x /3 for x [2/6, 3/6], g 3 (x) = 2x 5/6 for x [3/6, 5/6], x for x [5/6, ]. Taking g (x) = g 2 (x) = g 3 (x), this g 3 (x) can be used as a starting point for a further application of Theorem 4 which gives another solution of g f = g h. 3. Computing T (j+)/6 ( T (i+)/6 F (x, y) dx) dy for i, j =,..., 5 directly, j/6 i/6 we can find \ \ F (x, y) dg 3 (x) dg 3 (y) =, which is also a consequence of Theorem 3 and (g 3 ) f = (g 3 ) h. 4. Since the mapping g g φ is linear, the set of all solutions of g f = g h is convex. 5. Since x f = x h, Theorem 2 leads to the fact that the following distribution function g 4 (x) is not a distribution function of ξ(3/2) n mod, for any ξ R: g 4 (x) = { x for x [, 2/3], x 2 (2/3)x + 2/3 for x [2/3, ]. 6. By Figure, X = [2/9, /3] [/2, ] is also a set of uniqueness. Moreover, X = /8 < 2/3. Similarly for [, /2] [2/3, 7/9]. 7. Since all the components of f ([, x)) and ([, x)) are semiclosed the fact that, for fixed ξ and m, {ξ(3/2) m } = {ξ(3/2) n } only for finitely many n, was not used in the proof of Theorem. References [] M. Bennett, Fractional parts of powers of rational numbers, Math. Proc. Cambridge Philos. Soc. 4 (993), 9 2. [2] G. Choquet, Construction effective de suites (k(3/2) n ). Étude des mesures (3/2)- stables, C. R. Acad. Sci. Paris Sér. A-B 29 (98), A69 A74 (MR 82h:62a-d). [3] L. Flatto, J. C. Lagarias and A. D. Pollington, On the range of fractional parts {ξ(p/q) n }, Acta Arith. 7 (995), [4] H. Helson and J.-P. Kahane, A Fourier method in diophantine problems, J. Analyse Math. 5 (965), (MR 3#5856). [5] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 974. [6] K. Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8 (968), (MR 37#2694). [7] I. I. Piatetski-Shapiro, On the laws of distribution of the fractional parts of an exponential function, Izv. Akad. Nauk SSSR Ser. Mat. 5 (95), (MR 3, 23d) (in Russian).

11 Distribution functions of ξ(3/2) n mod 35 [8] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (959), [9] R. Tijdeman, Note on Mahler s 3/2-problem, Norske Vid. Selsk. Skr. 6 (972), 4. [] J. G. van der Corput, Verteilungsfunktionen I VIII, Proc. Akad. Amsterdam 38 (935), 83 82, 58 66; 39 (936), 9, 9 26, 49 53, , , [] T. Vijayaraghavan, On the fractional parts of the powers of a number, I, J. London Math. Soc. 5 (94), Mathematical Institute of the Slovak Academy of Sciences Štefánikova ul Bratislava, Slovakia strauch@mau.savba.sk Received on and in revised form on (298)