For , let be the self-similar set in generated by the iterated function system . In this paper, we investigate the intersection of the unit circle with the Cartesian product . We prove that for , the intersection is trivial, that is, If , then the intersection is nontrivial. In particular, if , the intersection is of cardinality continuum. Furthermore, the bound is sharp: there exists a sequence with such that is nontrivial for all . This result provides a negative answer to a problem posed by Yu (2023). Our methods extend beyond the unit circle and remain effective for many nonlinear curves. We also characterize the intersection of missing digits Cantor sets with the sequence by utilizing the Legendre symbol.

Given an integer M ⩾ 1 and β ∈ (1, M + 1), let Sβ,M be the fat Sierpinski gasket in ℝ2 generated by the iterated function system {f_d(x) = x+dβ : d ∈Ω_M} , where Ω_M = {(i, j) ∈ℤ_⩾0^2: i + j ⩽ M } . Then each x ∈ Sβ,M may be represented as a series x = ∑_i=1^∞d_iβ^i =: Π_β((d_i)) , and the infinite sequence (di) ∈ ΩMℕ is called a coding of x. Since β < M + 1, a point in Sβ,M may have multiple codings. Let Uβ,M be the set of x ∈ Sβ,M having a unique coding, i.e., U_β,M = {x ∈ S_β,M : #Π_β^-1(x) = 1 } . When M = 1, Kong and Li (2020) described two critical bases for the phase transitions of the intrinsic univoque set U_β,1 , which is a subset of Uβ,1. In this paper, we consider M ⩾ 2, and characterize the two critical bases βG(M) and βc(M) for the phase transitions of Uβ,M: (i) if β ∈ (1, βG(M)], then Uβ,M is finite; (ii) if β ∈ (βG(M), βc(M)), then Uβ,M is countably infinite; (iii) if β = βc(M), then Uβ,M is uncountable and has zero Hausdorff dimension; (iv) if β > βc(M), then Uβ,M has positive Hausdorff dimension. Moreover, we show that the first critical base βG(M) is a Perron number, while the second critical base βc(M) is a transcendental number.

In this paper, we study the analogous Erdős similarity conjecture in higher dimensions and generalize the Eigen-Falconer theorem. We show that if A={x_n}_n=1^∞⊆ℝ^d is a sequence of non-zero vectors satisfying lim_n →∞x_n =0 and lim_n →∞x_n+1/x_n = 1, then there exists a measurable set E ⊆ℝ^d with positive Lebesgue measure such that E contains no affine copies of A.

Given an integer $M\ge 1$ and $\beta\in(1, M+1)$, let $S_{\beta, M}$ be the fat Sierpinski gasket in $\mathbb R^2$ generated by the iterated function system $\left\{f_d(x)=\frac{x+d}{\beta}: d\in\Omega_M\right\}$, where $\Omega_M=\{(i,j)\in\mathbb Z_{\ge 0}^2: i+j\le M\}$. Then each $x\in S_{\beta, M}$ may represent as a series $x=\sum_{i=1}^\infty\frac{d_i}{\beta^i}=:\Pi_\beta((d_i))$, and the infinite sequence $(d_i)\in\Omega_M^{\mathbb N}$ is called a \emph{coding} of $x$. Since $\beta\beta_c(M)$ then $U_{\beta, M}$ has positive Hausdorff dimension. Our results can also be applied to the intrinsic univoque set $\widetilde{U}_{\beta, M}$. Moreover, we show that the first critical base $\beta_G(M)$ is a perron number, while the second critical base $\beta_c(M)$ is a transcendental number.

We introduce the generalized upper box dimension which is defined for any set, whether the set is bounded or unbounded. We study basic properties of the generalized upper box dimension. We prove that the generalized upper box and upper box dimensions coincide for bounded sets. Furthermore, we also show that the modified generalized upper box dimension equals the packing dimension. So the generalized upper box dimension can be seen as a reasonable generalization of the upper box dimension. As an application, we prove the generalized upper box dimension is zero if and only if the quasi-Assouad dimension is zero. We also show that the upper spectrum is of full dimension is equivalent to the Assouad spectrum is of full dimension and the upper spectrum is zero is equivalent to the Assouad spectrum is zero.

For λ∈(0,1/2) let K_λ be the self-similar set in ℝ generated by the iterated function system {f_0(x)=λx, f_1(x)=λx+1-λ}. In this paper, we investigate the intersection of the unit circle 𝕊⊂ℝ^2 with the Cartesian product K_λ × K_λ. We prove that for λ∈(0, 2 - √(3)], the intersection is trivial, i.e., 𝕊∩ (K_λ × K_λ) = {(0,1), (1,0)}. If λ∈ [0.330384,1/2), then the intersection 𝕊∩ (K_λ × K_λ) is non-trivial. In particular, if λ∈ [0.407493 , 1/2) the intersection 𝕊∩ (K_λ × K_λ) is of cardinality continuum. Furthermore, the bound 2 - √(3) is sharp: there exists a sequence {λ_n}_n ∈ℕ with λ_n ↘ 2 - √(3) such that 𝕊∩ (K_λ_n× K_λ_n) is non-trivial for all n∈ℕ. This result provides a negative answer to a problem posed by Yu (2023). Our methods extend beyond the unit circle and remain effective for many nonlinear curves. By employing tools from number theory, including the quadratic reciprocity law, we analyze the intersection of Cantor sets with some sequences. A dichotomy is established in terms of the Legendre symbol associated with the digit set, revealing a fundamental arithmetic constraint governing such intersections.

Let K be an imaginary quadratic field and let 𝒪_K be the ring of algebraic integers of K. For α∈𝒪_K with |α| > 1, define 𝒟_α= ⋃_n=0^∞𝒪_K/α^n. For β∈𝒪_K with |β|>1 and a finite subset A ⊂𝒪_K, define S_β,A = {∑_k=1^∞a_k/β^k: a_k ∈ A ∀ k ∈ℕ}. Suppose that α and β are relatively prime. In this paper, we show that if _H S_β,A < 1, then the intersection 𝒟_α∩ S_β,A is a finite set. In general, the threshold for the Hausdorff dimension of S_β,A is sharp. If we further assume that 𝒪_K is a unique factorization domain and that and α are relatively prime, then we establish the finiteness of the intersection under the weaker condition _H S_β,A < 2. This extends the previously known results on the real line.

We establish the pointwise equidistribution of self-similar measures in the complex plane. Let $\beta \in \mathbb Z[\mathrm{i}]$, whose complex conjugate $\overline{\beta}$ is not a divisor of beta, and $T \subset \mathbb Z[\mathrm{i}]$ a finite subset. Let mu be a non-atomic self-similar measure with respect to the IFS $\big\{f_{t}(z)=\frac{z+t}{\beta}\colon t\in T\big\}$. For $\alpha \in \mathbb Z[\mathrm{i}]$, if alpha and beta are relatively prime, then we show that the sequence $(\alpha<^>n z)_{n\ge 1}$ is equidistributed modulo one for mu -almost everywhere $z \in \mathbb{C}$. We also discuss normality of radix expansions in Gaussian integer base, and obtain pointwise normality. Our results generalize partially the classical results in the real line to the complex plane.

Given an integer M≥ 1 and β∈(1, M+1), let S_β, M be the fat Sierpinski gasket in ℝ^2 generated by the iterated function system {f_d(x)=x+d/β: d_M}, where Ω_M={(i,j)∈ℤ_≥ 0^2: i+j≤ M}. Then each x∈ S_β, M may be represented as a series x=∑_i=1^∞d_i/β^i=:Π_β((d_i)), and the infinite sequence (d_i)_M^ℕ is called a coding of x. Since ββ_c(M) then U_β, M has positive Hausdorff dimension. Our results can also be applied to the intrinsic univoque set U_β, M. Moreover, we show that the first critical base β_G(M) is a Perron number, while the second critical base β_c(M) is a transcendental number.

Fix a positive integer N and a real number 0< β < 1/(N+1) . Let Γ be the homogeneous symmetric Cantor set generated by the IFS {ϕ _i(x)=β x + i 1-β/N: i=0,1,… , N }. For m∈ℤ_+ we show that there exist infinitely many translation vectors t=(t_0,t_1,… , t_m) with 0=t_0

For$\lambda \in (0,\,1/2]$let$K_\lambda \subset \mathbb {R}$be a self-similar set generated by the iterated function system$\{\lambda x,\, \lambda x+1-\lambda \}$. Given$x\in (0,\,1/2)$, let$\Lambda (x)$be the set of$\lambda \in (0,\,1/2]$such that$x\in K_\lambda$. In this paper we show that$\Lambda (x)$is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any$y_1,\,\ldots,\, y_p\in (0,\,1/2)$there exists a full Hausdorff dimensional set of$\lambda \in (0,\,1/2]$such that$y_1,\,\ldots,\, y_p \in K_\lambda$.

In this paper, we study the spectrality of infinite convolutions in ℝ^d , where the spectrality means the corresponding square integrable function space admits a family of exponential functions as an orthonormal basis. Suppose that the infinite convolutions are generated by a sequence of admissible pairs in ℝ^d . We give two sufficient conditions for their spectrality by using the equi-positivity condition and the integral periodic zero set of Fourier transform. By applying these results, we show the spectrality of some specific infinite convolutions in ℝ^d .

In this paper, we explore spectral measures whose square integrable spaces admit a family of exponential functions as an orthonormal basis. Our approach involves utilizing the integral periodic zeros set of Fourier transform to characterize spectrality of infinite convolutions generated by a sequence of admissible pairs. Then we delve into the analysis of the integral periodic zeros set. Finally, we show that given finitely many admissible pairs, almost all random convolutions are spectral measures. Moreover, we give a complete characterization of spectrality of random convolutions in some special cases. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let { ( N j , B j , L j ) : 1 ⩽ j ⩽ m } be finitely many Hadamard triples in R . Given a sequence of positive integers { n k } k = 1 ∞ and ω = ( ω k ) k = 1 ∞ ∈ { 1 , 2 , … , m } N , let μ ω , { n k } be the infinite convolution given by μ ω , n k = δ N ω 1 − n 1 B ω 1 ∗ δ N ω 1 − n 1 N ω 2 − n 2 B ω 2 ∗ ⋯ ∗ δ N ω 1 − n 1 N ω 2 − n 2 ⋯ N ω k − n k B ω k ∗ ⋯ . In order to study the spectrality of μ ω , { n k } , we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if g c d ( B j − B j ) = 1 for 1 ⩽ j ⩽ m , then all infinite convolutions μ ω , { n k } are spectral measures. This implies that we may find a subset Λ ω , { n k } ⊆ R such that { e λ ( x ) = e 2 π i λ x : λ ∈ Λ ω , { n k } } forms an orthonormal basis for L 2 ( μ ω , { n k } ) .

We consider the iterated function system (IFS) f q → ( z → ) = z → + q → β , q → ∈ { ( 0 , 0 ) , ( 1 , 0 ) , ( 0 , 1 ) } . \begin{equation*} f_{\vec {q}}(\vec {z})=\frac {\vec {z}+\vec {q}}{\beta },\,\,\vec {q}\in \{(0,0),(1,0),(0,1)\}. \end{equation*} As is well known, for β = 2 \beta = 2 the attractor, S β S_\beta , is a fractal called the Sierpiński gasket (or sieve) and for β > 2 \beta >2 it is also a fractal. Our goal is to study random β \beta -transformations on the attractor for this IFS with 1 > β ≤ 3 / 2 1>\beta \leq 3/2 . In this case, S β S_\beta is a triangle. We show that all β \beta -expansions of a point z → \vec {z} in S β S_\beta can be generated by a random map K β K_\beta defined on { 0 , 1 } N × { 0 , 1 , 2 } N × S β \{0,1\}^\mathbb {N}\times \{0,1,2\}^\mathbb {N}\times S_\beta and K β K_\beta has a unique invariant measure of maximal entropy. Furthermore, we show the existence of a K β K_\beta -invariant probability measure of the form m 1 ⊗ m 2 ⊗ μ β m_1\otimes m_2 \otimes \mu _{\beta } , where m 1 , m 2 m_1, m_2 are product measures on { 0 , 1 } N , { 0 , 1 , 2 } N \{0,1\}^\mathbb {N},\{0,1,2\}^\mathbb {N} , respectively, and μ β \mu _{\beta } is absolutely continuous with respect to the two-dimensional Lebesgue measure λ 2 \lambda _2 .

Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~ \forall i\in\mathbb{N} \bigg\}, $$ where $\mathcal{A} \subset \{0,1,\ldots, q-1\}$ with cardinality $1<\#\mathcal{A}< q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,\mathcal{A}))<+\infty$. In this paper we show that for any $r\in\mathbb{Q}$ and for any $\alpha\in\mathbb{R}$, $$ \#\big((r D_p+\alpha)\cap K(q,\mathcal{A})\big)<+\infty. $$

Given β ∈ (0 , 1 / 3), let Γ be the middle-(1 − 2 β ) Cantor set. In this paper we give a complete characterization on which the union S mj =0 (Γ + t j ) is a self-similar set. Furthermore, for any m ∈ N we show that there exist infinitely many translation vectors t = ( t 0 , t 1 , . . . , t m ) with 0 = t 0 < t 1 < · · · < t m such that the union S mj =0 (Γ + t j ) is a self-similar set.

We introduce a class of sets defined by digit restrictions in [Formula: see text] and study its fractal dimensions. Let [Formula: see text] be a set defined by digit restrictions in [Formula: see text]. We obtain the Hausdorff and lower box dimensions of [Formula: see text]. Under some condition, we gain the packing and upper box dimensions of [Formula: see text]. We get the Assouad dimension of [Formula: see text] and show that it is 2 if and only if [Formula: see text] contains arbitrarily large arithmetic patches. Under some conditions, we study the upper spectrum, quasi-Assouad dimension and Assouad spectrum of [Formula: see text]. Finally, we give an intermediate value property of fractal dimensions of the class of sets.

Given a positive integer m, let Omega(m) = {0, 1,..., m}, and let B-2(m) denote the set of bases q is an element of (1, m+ 1] in which there exist numbers having precisely two q-expansions over the alphabet Omega m. Sidorov [23] firstly studied the set B2(1) and raised some questions. Komornik and Kong [15] further investigated the set B-2(1) and partially answered Sidorov's questions. In the present paper, we consider the set B-2(m) for general positive integer m, and generalise the results obtained by Komornik and Kong.