The achievement set of generalized multigeometric sequences

We study the topology of all possible subsums of the generalized multigeometric series $k_1f(x)+k_2f(x)+\dots+k_mf(x)+\dots + k_1f(x^n)+\dots+k_mf(x^n)+\dots,$ where $k_1, k_2, \dots, k_m$ are fixed positive real numbers and $f$ runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval.


Introduction
Let E(z n ) denote the set of all possible sums of elements of the sequence {z n }, or equivalently, all possible subsums of the series n≥1 z n .That is, Also known as the achievement set of the sequence {z n } [7,9,11], it was first considered by Kakeya [8] who conjectured that, for a convergent positive series, this set is either a finite union of compact intervals or a Cantor set.This conjecture was refuted in [5] by means of the following result whose proof was completed in [13].
Theorem 1.1.The achievement set of a summable positive sequence is either: (i) a finite union of closed bounded intervals, (ii) homeomorphic to the Cantor set, or (iii) homeomorphic to a Cantorval.
Recall that a (symmetric) Cantorval can be formally defined as a nonempty compact real subspace which is the closure of its interior and the endpoints of any nontrivial component of this set are accumulation points of trivial components.All Cantorvals are homeomorphic to [0, 1]\ ∪ n≥1 B 2n in which B n is the union of the 2 n−1 open intervals which are eliminated at the nth stage of the construction of the Cantor set, which can be seen as E( 23 n ).In particular, any Cantorval is homeomorphic to the Guthrie-Nymann Cantorval given by E(z n ), with z 2n = 2/4 n and z 2n−1 = 3/4 n , the one originally considered in (iii) of the above Theorem.Cantorvals also appear as attractors associated to some iterated function systems [2], and an analogous result to Theorem 1.1 holds to describe the topology of the algebraic difference of certain Cantor sets [1,10,12].
In this paper we consider a class of positive functions f defined on the interval (see next section) and study the topological behavior, depending on x, of the set all possible subsums of the series where k 1 , . . ., k m are fixed positive scalars.Following the nomenclature in [4] we call this a generalized multigeometric series.We show that, whenever x varies along some particular regions, the achievement set of the associated sequence is a compact interval (Theorem 2.8 and Corollary 2.9), a Cantor set (Theorem 2.14), or a Cantorval (Corollary 2.12).This extends the main results of [14] where f (x) = sin(x), and those in [3] and [4] where f (x) = x, i.e., the considered series is the multigeometric, where k 1 , k 2 , . . .k m are fixed positive integers and q ∈ (0, 1).
Finally, notice that the achievement set for the sequence associated to the multigeometric series (1.2) equals the arithmetic sum of the achievement sets associated to the different multigeometric series given by any partition of the scalars k 1 , . . ., k m .Therefore the results here can also be seen as criteria to determine the topological nature of the arithmetic sum of the achievement sets of the multigeometric sequences.

The achievement set of some generalized multigeometric sequences
Observe that the generalized multigeometric series given in (1.2), associated to a given function f and to positive real numbers k 1 , . . ., k m , can be written as where, as usual, ⌊ • ⌋ denotes the integer part function and Observe also that the achievement set for this sequence at a given x is We will study the topology of this set for a particular class of functions: Definition 2.2.A function f is locally increasing and power bounded (at 0) if there exist ǫ ∈ (0, 1), and a, b, r ∈ R + such that f is monotone increasing in [0, ǫ] and We denote by M the class of locally increasing at 0 and power bounded functions.
The following shows that differentiable functions abound in M: Proof.With f as in the statement there exists ǫ ∈ (0, 1) such that f r) ([0, ǫ]) ⊂ R + , and therefore f (x) is monotone increasing in [0, ǫ].
On the other hand, define and consider, for any λ ∈ R, the function h λ (x) = f (x) − λx r .We then use the Taylor (r − 1)th-approximation together with the error formula of h λ at 0 to conclude that (2.4) In particular, for any (2) By the well known Jordan inequality, we see that the function We also need: Proof.The second assertion trivially holds: as f is monotone increasing in [0, ǫ] it follows that, for any x in this interval, f On the other hand, if we write K = m i=1 k i , we deduce that and therefore, this series converges since ǫ ∈ (0, 1).
In what follows we fix an arbitrary function f ∈ M, thus constants ǫ ∈ (0, 1], and a, b, r ∈ R + are those given in Definition 2.2.We choose positive real numbers k 1 ≥ k 2 ≥ • • • ≥ k m > 0, and consider the associated generalized multigeometric series n≥1 w n (x) for x ∈ [0, ǫ).We also fix the following notation: Also, for any series n≥1 z n and any ℓ ≥ 1 we denote by Z ℓ = n>ℓ z n the ℓth tail of the series.In particular, we write W ℓ (x) = n>ℓ w n (x).
On the other hand, we will strongly use the following foundational result of Kakeya Proof.According to the Theorem 2.7.(ii), it is enough to show that w n (x) ≤ W n (x) for every n ∈ N and any x ∈ [d I , ǫ).Since f ∈ M, For x = 0 this trivially holds and, for x > 0 this is the case when for all n.As the function g(n) takes the values 1, 2, . . ., m, this happens for all x ∈ [d I , ǫ).
As a consequence, denoting we obtain: Proof.Simply note that and apply Theorem 2.8.
On the other hand, denoting we prove the following that extends [14, Theorem 3.3], which in turn is inspired by [3, Theorem Theorem 2.10.The achievement set E w n (x) is not a finite union of closed bounded intervals for 0 < x < min{ǫ, d N I }.
Proof.According to Theorem 2.7.(i) it is sufficient to show that w ℓm (x) > W ℓm (x) for any ℓ ∈ N and 0 < x < min{ǫ, d N I }.Observe that, for any x ∈ (0, ǫ), Therefore w ℓm (x) > W ℓm (x) as long as The next result extends [3, Theorem 2.1], [4, Theorem 2.2(i)] and [14,Theorem 3.4].We follow a strategy similar to the proofs of these references.We finish the proof by showing that the interval where s n ∈ {0, 1, . . ., s}.Therefore, there exist c n,i ∈ {0, 1} such that µ The following recovers [4, Theorem 2.2(iii)] and generalizes [14,Corollary 3.5].Under the hypothesis of the previous theorem we have: x < r akm akm+bK , then w mi (x) > W mi (x) for every i ∈ N and E(w n ) cannot be a finite union of closed and bounded intervals.
Therefore, E w n (x) must be a Cantorval.for all n and the theorem follows.

Example 2 . 5 .
(1) Note that the identity f (x) = x is trivially in M by choosing a = b = r = 1 and any ǫ.

Corollary 2 . 12 .
Whenever d CI < d N I , the achievement set E w n (x) is a Cantorval for any x ∈ [d CI , d N I ).Proof.Let x ∈ [d CI , d N I ).Then (i) According to the Theorem 2.11, if x ≥ r b sa+b , then E w n (x) contains an interval.(ii) An analogous argument to the one in the proof of Theorem 2.10 shows that if

Remark 2 . 13 .Theorem 2 . 14 .
Observe that d CI < d N I only if b < a sk m K .Therefore, as a ≤ b, 1 < sk m K is a necessary condition for Corollary 2.12.Finally, letd C = r min 1≤j≤m ak j − bU j ak j + bL j .Our last result extends and refines[14, Theorem 3.7]:Whenever d C > 0, the achievement set E w n (x) is homeomorphic to the Cantor set for 0 < x < min{ǫ, d C }.Proof.According to Theorem 2.7.(i), it is enough to show that w n (x) > W n (x) for all but finitely many n ∈ N and x ∈ [0, d C ]. Observe that, for any x ∈ (0, ǫ),w n (x) = k g(n) f (x ⌊ m+n−1 m ⌋ ) ≥ a • k g(n) x ⌊ m+n−1 m ⌋r while W n (x) = ℓ>n w ℓ ≤ b • ℓ>n k g(ℓ) x ⌊ m+ℓ−1 m ⌋r = b • x ⌊ m+n−1 m ⌋r U g(n) + Kx r 1 − x r .Therefore, w n (x) > W n (x) as long asa • k g(n) x ⌊ m+n−1 m ⌋r > b • x ⌊ m+n−1 m ⌋r U g(n) + Kx r 1 − x r , and since x > 0, whenever a • k g(n) > b • U g(n) + Kx r 1 − x r .That is, when x < r ak g(n) − bU g(n)ak g(n) + bL g(n) Let n≥1 z n be a convergent positive series with non-increasing terms, i.e., z n ≥ z n+1 for any n ∈ N.Then, the achievement set E(z n ) is:(i) a finiteunion of bounded closed intervals if and only if z n ≤ Z n for all but finitely many n ∈ N; (ii) a compact interval if and only if z n ≤ Z n for every n ∈ N; (iii) homeomorphic to the Cantor set if z n > Z n for all but finitely many n ∈ N.
1≤j≤m bk j − aU j bk j + aL j .Our first result extends and refines [14, Theorem 3.1]: Theorem 2.8.Whenever d I ≤ ǫ, the achievement set E w n (x) is a compact interval for any x ∈ [d I , ǫ].