Add to ORKGGiven x ∈ (0, 1], let U(x) be the set of bases q ∈ (1, 2] for which there exists a unique sequence (di ) of zeros and ones such that x = ∑∞ i=1 di /q i . Lü et al. (2014) proved that U(x) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U(x) + λU(x) and product U(x) · U(x) λ contain an interval for all x ∈ (0, 1] and λ ̸= 0. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (Dajani and Kalle, 2017)
Let 1<β<2. Given any x∈[0,(β−1)−1], a sequence (an)∈{0,1}N is called a β-expansion of x if x=∑∞n=1an...
In this paper, we are concerned with two exceptional sets arising in the Pierce expansion of numbers...
This paper is devoted to providing a unifying approach to the study of the uniqueness of uncondition...
AbstractErdős, Horváth and Joó discovered some years ago that for some real numbers 1<q<2 there exis...
Erdos, Horvath and Joo discovered some years ago that for some real numbers 1 < q < 2 there exists o...
In this note we will show that for every natural number n> 0 there exists an S ⊂ [0, 1] such that...
Abstract. We present a theorem which generalizes some known theorems on the existence of nonmeasurab...
We show that every unconditional basis in a finite direct sum ⊕p∈Aℓp , with A ⊂ (0,∞], splits into u...
Given a positive integer M and a real number q∈(1,M+1], an expansion of a real number x∈[0,M/(q−1)] ...
We prove that there exist positive constants C and c such that for any integer d⩾ 2 the set o...
AbstractLet m, n, be positive integers and let Q be an infinite subset of Zn. For any number τ, defi...
Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we conside...
Given a positive integer M and q∈(1,M+1], let Uq be the set of x∈[0,M/(q−1)] having a unique q-expan...
We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X h...
AbstractIt is shown that the arithmetic sum of middle-α Cantor sets typically has positive Lebesgue ...
Let 1<β<2. Given any x∈[0,(β−1)−1], a sequence (an)∈{0,1}N is called a β-expansion of x if x=∑∞n=1an...
In this paper, we are concerned with two exceptional sets arising in the Pierce expansion of numbers...
This paper is devoted to providing a unifying approach to the study of the uniqueness of uncondition...
AbstractErdős, Horváth and Joó discovered some years ago that for some real numbers 1<q<2 there exis...
Erdos, Horvath and Joo discovered some years ago that for some real numbers 1 < q < 2 there exists o...
In this note we will show that for every natural number n> 0 there exists an S ⊂ [0, 1] such that...
Abstract. We present a theorem which generalizes some known theorems on the existence of nonmeasurab...
We show that every unconditional basis in a finite direct sum ⊕p∈Aℓp , with A ⊂ (0,∞], splits into u...
Given a positive integer M and a real number q∈(1,M+1], an expansion of a real number x∈[0,M/(q−1)] ...
We prove that there exist positive constants C and c such that for any integer d⩾ 2 the set o...
AbstractLet m, n, be positive integers and let Q be an infinite subset of Zn. For any number τ, defi...
Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we conside...
Given a positive integer M and q∈(1,M+1], let Uq be the set of x∈[0,M/(q−1)] having a unique q-expan...
We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X h...
AbstractIt is shown that the arithmetic sum of middle-α Cantor sets typically has positive Lebesgue ...
Let 1<β<2. Given any x∈[0,(β−1)−1], a sequence (an)∈{0,1}N is called a β-expansion of x if x=∑∞n=1an...
In this paper, we are concerned with two exceptional sets arising in the Pierce expansion of numbers...
This paper is devoted to providing a unifying approach to the study of the uniqueness of uncondition...