More on Kakeya Conditions for Achievement Sets

We identify an incorrect estimate in the proof of one of principal theorems from Marchwicki and Miska (Results Math, 2021. https://doi.org/10.1007/s00025-021-01479-2) and demonstrate that the original construction of a special series with unique subsums remains valid when using a weaker estimate that we prove to be true. Additionally, we present a weaker version—without the uniqueness of subsums—of the Thm. 2.1 from Marchwicki and Miska (2021), but with a very simple proof based on the concept of semi-fast convergent series.

The main goal of this short note is to expose an incorrect estimate in the intricate proof of one of the principal theorems of the recently published paper [2]. The proof is based on a very delicate and complicated construction of a special convergent series and we have decided to write this note as an extension of the forementioned paper, assuming in particular that the reader will be familiar with all preliminaries and notation used in the paper [2] which allows us to use identical notation and definitions without adding any preliminaries to this note. The theorem whose proof requires an amendment is the following [2, Thm. 2.1]. Theorem 1. Let C = {n 1 < n 2 < n 3 < · · · } be an infinite subset of N. Then there exists a non-increasing sequence (x n ) of positive real numbers such that the series ∞ n=1 x n is convergent, {n ∈ N : x n > r n } = C, {n ∈ N : x n < r n } = C c and A(x n ) = U (x n ). In particular, A(x n ) is a Cantor set. The first part of the original proof [2, pp. 4-6] is devoted to the definition an auxiliary sequence (y n ), the main sequence (x n ) and to some crucial properties of these sequences. The second part of the proof [2, pp. 6-7] deals with uniqueness of subsums which is then used to conclude that the achievement set A(x n ) necessarily is a Cantor set. However, it came to our attention that one estimate in the second part is incorrect. Namely, in the second line from the top on the seventh page of the article the Authors claim that However, under the assumptions (1), we have for any m between 1 and and hence, in the case (1), could fail in the case (1) when n k+1 − n k > 5. Fortunately, the original proof can be rescued by observing that the equality (2) yields-in the discussed subcase-the following estimate and hence (see [2], lines 6-9 from the top of the page 7) n∈F ∩I k+1 and the last inequality is exactly(!) the inequality 2 n k−1 −n k y k > r n k+1 + y k+1 (with k replaced by k + 1) that was proved on page 5 of [2].
We conclude this note with a weak version of the Thm. 2.1 from [2] that, firstly, preserves the crucial part of the thesis of the theorem and, secondly, admits a rather simple proof based on the omission of the argument of A(x n ) = U (x n ) and on the use of the concept of semi-fast convergent series instead.
A series x n with monotonic and positive terms convergent to 0 is called semi-fast convergent [1] if it satisfies the condition

If
x n is a semi-fast convergent series, then there exist two uniquely determined sequences, (α k ) of positive numbers decreasing to 0 and (N k ) of positive integers such that where we put N 0 := 0. The numbers α k are the values of the terms of the series x n and N k is the multiplicity of the value α k in the series. Thus, we can identify and the sum of the series is Below is the weaker version of the Thm. 2.1 from [2] that can be proven by means of semi-fast convergent series. Theorem 2. Let C = {n 1 < n 2 < n 3 < . . .} be an infinite subset of N. Then there is a non-increasing sequence (x n ) of positive real numbers such that the series ∞ n=1 x n converges, {n ∈ N : x n > r n } = C, {n ∈ N : x n < r n } = C c and A(x n ) is a Cantor set.
Proof. Define n 0 := 0 and choose any x 0 > 0. Further, let I k := {n k−1 + 1, n k−1 + 2, . . . , n k } for k ∈ N. Then we define by induction a sequence (x i ) i∈N such that Then for all k i∈I k and hence by induction on m ∀m ∈ N i∈I k+m The following estimate holds for the n k -th remainder of the series ∞ i=1 x i : Thus (x i ) i∈N = (α k , N k ) k∈N with α k = x n k and N k = n k − n k−1 , that is, ∞ i=1 x i is a semi-fast convergent series and hence A(x i ) is a Cantor set by the Thm. 16 from [1].
Clearly, the equalities {n ∈ N : x n > r n } = C, {n ∈ N : x n < r n } = C c hold for the series.