Related papers: Finite-State Dimension and Real Arithmetic

For each odd prime power q, and each integer k, we determine the sum of the k-th powers of all elements x in F_q for which both x and x+1 are squares in F_q^*. We also solve the analogous problem when one or both of x and x+1 is a…

We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal…

In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students…

Let $b \geq 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any non-dense set $A \subset [0,1)$ such that $A \cap \mathbb{Q}$ is invariant under $\times b$ operation, we prove the finiteness…

We study dynamical systems with the property that all the nontrivial factors have infinite topological entropy (or, positive mean dimension). We establish an ``if and only if'' condition for this property among a typical class of dynamical…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q…

Let $\mathbb F$ denote an algebraically closed field and assume that $q\in \mathbb F$ is a primitive $d^{\rm \, th}$ root of unity with $d\not=1,2,4$. The universal Askey--Wilson algebra $\triangle_q$ is a unital associative $\mathbb…

We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that,…

We study the generalized Hausdorff dimension of some natural subsets of $k^{-1}(3)$, where $k^{-1}(3)$ consists of the real numbers $x$ for which $\left| x-\frac{p}{q} \right|<\frac{1}{(3+\varepsilon)q^2}$ has infinitely many rational…

We study the topological, dynamical, and descriptive set theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian…

We observe that a finitely generated algebraic algebra R (over a field) is finite dimensional if and only if the associated graded ring grR is right noetherian, if and only if grR has right Krull dimension, if and only if grR satisfies a…

Given a prime $p>3$, we characterize positive-definite integral quadratic forms that are coprime-universal for $p$, i.e. representing all positive integers coprime to $p$. This generalizes the $290$-Theorem by Bhargava and Hanke and extends…

We derive an inequality for the linear entropy, that gives sharp bounds for all finite dimensional systems. The derivation is based on generalised Bloch decompositions and provides a strict improvement for the possible distribution of…

This work is devoted to proving that, given an integer $x \ge 2$, there are infinitely many perfect powers, coprime with $x$, having exactly $k \ge 3$ non-zero digits in their base $x$ representation, except for the case $x=2, k=4$, for…

We define an extension of parity from the integers to the rational numbers. Three parity classes are found -- even, odd and `none'. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The…

We prove the consistency of the existence of a $Q$-set whose square is not a $\Delta$-set and that if there is a $\Delta$-set, then there exists a $\Delta$-set whose all finite powers are $\Delta$-sets.

When $A$ and $B$ are subsets of the integers in $[1,X]$ and $[1,Y]$ respectively, with $|A| \geq \alpha X$ and $|B| \geq \beta X$, we show that the number of rational numbers expressible as $a/b$ with $(a,b)$ in $A \times B$ is $\gg (\alpha…

Kronecker sequences $(k \alpha \mod 1)_{k=1}^{\infty}$ for some irrational $\alpha > 0$ have played an important role in many areas of mathematics. It is possible to associate to each finite segment $(k \alpha \mod 1)_{k=1}^{n}$ a…

Since E. Borel proved in 1909 that almost all real numbers with respect to Lebesgue measure are normal to all bases, an open problem has been whether simple irrationals like square root of 2 are normal to any base. We show that each number…