Uniform distribution theory Feed

Ramsey-Collatz Correlation and Some Extremal Combinatorics, along with Ramsey-Mahler Rare Concurrencies

Wed, 31 Dec 2025 00:00:00 GMT

For a ternary tree T of depth n with 0-1 labeled edges, its weight f (T ) is the least number of path labels among binary subtrees. The maximum f (n), over all labelings, of these weights starts with 1,2,3,4,8. We show f (6) ≥ 12, but focus on depth 5 (with 2363 trees). We approximate the percentages for weights 1–8: 0, 1.04, 23.6, 55.0, 18.8, 1.54, 0, 0; our linked supplements include thousands of mined trees of rare weights 7-8. Our next products additionally relate to Mahler’s 32 {3 \over 2} -problem and large stopping times showing a real number is not a Z-number. Our version is iterated multiplication of integers by 23 {2 \over 3} . For a certain sequence of integer intervals, we present a choice function. Our interval Ig has left endpoint a(g)=min{k|{g⋅(23)k}≥12∨k=⌈log32(g)⌉} a(g) = \min \left\{ k \mid \left\{ g \cdot \left(\frac{2}{3}\right)^k \right\} \ge \frac{1}{2} \;\vee\; k = \left\lceil \log_{\frac{3}{2}}(g) \right\rceil \right\} , and right endpoint at the stopping time for “the 1st intermediate rounding error”, but the interpolation is a “no sudden death” function. We present simultaneous peculiarity in base 2 and base 3: numbers g with large values (for the size of g)of a(g) and length of Ig, which also have a less common Ramsey weight (when written in base 2 and used to edge-label a tree), or are prime. Then we cross the weight notion with the Collatz scaled total stopping time γ∞(n). We construct sizable low-high sequences of 8-tuples of same-weight pairs of numbers below 2363 with certain monotonicity in values of γ∞, and in six apartness levels > 10−i for i =1,..., 6. Apartness of γ∞-values would be in the ‘low’ and ‘high’ halves as well as between the corresponding components of terms of the sequence. We get lower bounds for their lengths, and for Collatz-landing-apart just for weight 8. We statistically establish an unexpected correlation between Collatz and Ramsey.

Convergence of Multiple Series and ℐ–Variation

Wed, 31 Dec 2025 00:00:00 GMT

In the classical theory of multiple series the notion of variation of a sequence plays an important role. According to the concept of ℐ−convergence the notion of ℐ−variation of a sequence was recently introduced. In this paper we will examine applications of ℐ−variation of a sequence mainly in the theory of multiple series in particular in relation with the notion of ℐ−convergence.

Arithmetic Independence of Certain Uniform Sets of Algebraic Integers

Wed, 31 Dec 2025 00:00:00 GMT

We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetic independence. Here, “almost uniformity” means that the elements of a finite set are distributed lmost equidistantly in the unit interval, while “arithmetic independence” means that the number fields generated by the elements of a set do not have a mutual inclusion relation each other. Furthermore, we reveal to what extent the algebraic number fields generated by the elements of the four sets can cover quadratic or cubic fields.

On the Lipschitz Continuity of the Spherical Cap Discrepancy around Generic Point Sets

Wed, 31 Dec 2025 00:00:00 GMT

The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy.

Temporal Central Limit Theorem for a Multidimensional Adding Machine

Wed, 31 Dec 2025 00:00:00 GMT

Let p1,...,ps+1 be distinct primes and let Tpi be the von Neumann-Kakutani adding machine (1 ≤ i ≤ s), T𝒫 (x)= (Tp1 (x1),...,Tps(xs)). Let yi ∈ (0, 1) be a ps+1-rational (1 ≤ i ≤ s), 𝟙[0,y) the indicator function of the box [0,y1)×…×[0,ys). In this paper, we prove the following central limit theorem: ∑k=-nn-1𝟙[0,y)(TPk(x)-2ny1y2⋯ys)ℋN(x)⋅log2s/2N→N(0,1) \frac{\sum_{k=-n}^{\,n-1}\mathbb{1}_{[0,y)}\!\left(T_{P}^{k}(\mathbf{x})- 2n\,y_{1}y_{2}\cdots y_{s}\right)}{\mathcal{H}_{N}(\mathbf{x})\,\log_{2}^{\,s/2} N}\xrightarrow{w} \mathcal{N}(0,1) when n is sampled uniformly from {1,...,N}, ℋN (x)∈[υ1,υ2] with some υ1,υ2 > 0, for almost all x ∈[0, 1)s. The main tool in the proof is the S-unit theorem and the theorem on linear forms in the logarithm.

Low Discrepancy Digital Kronecker-Van der Corput Sequences

Wed, 31 Dec 2025 00:00:00 GMT

The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number d, any collection of one dimensional so-called low discrepancy sequences {Si : 1 ≤ i ≤ d} can be concatenated to create a d-dimensional hybrid sequence (S1,...,Sd). Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. The purpose of this note is to remedy this by providing an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences.Specifically, let 𝔽q be the finite field of cardinality q. It is shown that some real numbered hybrid sequence H(Θ(t), P (t)) := H(Θ, P ) built from the digital Kronecker sequence associated to a Laurent series Θ(t) ∈ 𝔽q((t−1)) and the digital Van der Corput sequence associated to an irreducible polynomial P (t) ∈ 𝔽q[t] meets the above property. More precisely, if Θ(t)is a counterexample to the so called t-adic Littlewood Conjecture (t-LC), then another Laurent series Φ(t) ∈ 𝔽q((t−1)) induced from Θ(t)and P (t) can be constructed so that H(Φ, P ) is low discrepancy. Such counterexamples to t-LC are known over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the author.

Distribution of Leading Digits of Imaginary Parts of Riemann Zeta Zeros II

Wed, 31 Dec 2025 00:00:00 GMT

Let γn, n = 1,2, …, be the sequence of all positive imaginary parts of non-trivial zeros of the Riemann zeta function 𝜁(s) in ascending order. n this paper we give a quantitative result on the distribution of leading block of digits of γn, n = 1,2, …. For Ni =# {n ≥ 1: γn < 10i}(≥ 2) and an r-digits number D = d1d2 ··· dr, we show that the frequency. #{ n≤Ni:(leading block of r digits of γn )=D }Ni {{\# \left\{ {n \le {N_i}:\left( {\text{leading block of} \,r\,\text{digits of}\,{\gamma _n}\,} \right) = D} \right\}} \over {{N_i}}} is approximated by 19⋅10r-1+log1018π10r-1((D+1)log10(D+1)-Dlog10D-r-19)10i-10Ni {1 \over {9 \cdot {{10}^{r - 1}}}} + {{\log 10} \over {18\pi {{10}^{r - 1}}}}\left( {\left( {D + 1} \right){{\log }_{10}}\left( {D + 1} \right) - D{{\log }_{10}}D - r - {1 \over 9}} \right){{{{10}^i} - 10} \over {{N_i}}} with the explicit error bound 0.516i2Ni+1.112ilogiNi+10.452iNi+1.668logiNi-1.4741Ni. 0.516{{{i^2}} \over {{N_i}}} + 1.112{{i\log i} \over {{N_i}}} + 10.452{i \over {{N_i}}} + 1.668{{\log i} \over {{N_i}}} - 1.474{1 \over {{N_i}}}.

An Erdös-Turan Inequality for Compact Simply Connected Semisimple Lie Groups

Wed, 31 Dec 2025 00:00:00 GMT

The classical Erdös-Turan Inequality bounds how far a sequence of points in the circle is from being equidistributed in terms of its exponential moments. We prove an analogous inequality for all compact simply-connected semisimple Lie groups, bounding how far a sequence is from being equidistributed in the conjugacy classes of the group in terms of the moments of irreducible characters.

A Note on Matrices Over ℤ with Entries Stemming from Binomial Coefficients and from Catalan Numbers Once Pure and Once Taken Modulo 2

Tue, 03 Jun 2025 00:00:00 GMT

The Pascal matrix, which is related to Pascal’s triangle, appears in many places in the theory of uniform distribution and in many other areas of mathematics. Examples are the construction of low-discrepancy sequences as well as normal numbers or the binomial transforms of Hankel matrices. Hankel matrices which are defined by Catalan numbers and related to the paperfolding sequence are interesting objects in number theory. Therefore, matrices that share many properties with the Pascal matrix or such Hankel matrices are of interest. In this note we will collect common features of the Pascal matrix and the same modulo 2 as well as the Hankel matrix defined by Catalan numbers once pure and once modulo 2 in the ring of integers. Hankel matrices with only 0 and 1 entries in e.g., finite fields gave recently access to counterexamples to the so-called X-adic Liouville conjecture. This justifies as well as motivates our consideration of further matrices with 0 and 1 entries.

Valuation of American Lookback-Options and the Distribution of Kronecker Sequences

Tue, 03 Jun 2025 00:00:00 GMT

In this paper, we determine the complexity of calculating the fair value of an American lookback-option in a binomial n–step model via backwardation. It will turn out, that the complexity also depends on diophantine properties of a certain parameter of the model. Therefore, we will need some basic tools from the theory of uniform distribution for the determination of the complexity.

A Family of Periodic Orbits in the Liénard Differential Systems

Tue, 03 Jun 2025 00:00:00 GMT

In this paper, we investigate the existence of a family of periodic orbits for the Liénard systems by expressing these systems as a Hamiltonian system with two degrees of freedom The main technique applied here is the averaging theory.

On the Expected ℒ2-Discrepancy of Stratified Samples from Parallel Lines

Mon, 24 Feb 2025 00:00:00 GMT

We study the expected ℒ2-discrepancy of stratified samples generated from special equi-volume partitions of the unit square. The partitions are defined via parallel lines that are all orthogonal to the diagonal of the square. It is shown that the expected discrepancy of stratified samples derived from these partitions is a factor 2 smaller than the expected discrepancy of the same number of i.i.d uniformly distributed random points in the unit square. We conjecture that this is best possible among all partitions generated from parallel lines.

Creating Normal Numbers Using the Prime Divisors of Consecutive Integers

Mon, 24 Feb 2025 00:00:00 GMT

For each integer n ≥ 2, let p1 ≤ p2 ≤ ··· ≤ pk be the complete list of the prime factors of a(n):= n(n+1). Consider the function sn : {p1,..., pk}→ {0, 1} defined by sn(pj)= 0 if pj | n and 1 if pj | n + 1. Then consider the binary number h(n) := sn(p1) ...sn(pk). In an earlier paper, we proved that the number 0.h(2) h(3) h(4) ... is a binary normal number and in fact we proved the more general statement when, for a fixed integer t ≥ 2, we set a(n) := n(n +1) ··· (n + t − 1), thus allowing for the construction of a normal number in base t. Here, we give a much shorter and simpler proof of this result and then we consider a more general result when a(n) is the product of linear functions.

On Jordan Double Sums and Related Summatory Functions

Mon, 24 Feb 2025 00:00:00 GMT

We study the Jordan double sums ∑[ d,δ ]≤NJɛ(d)Jɛ(δ)[ d,δ ]σ+2ɛ+it, \sum\limits_{\left[ {d,\delta } \right] \le N} {{{{J_\varepsilon }\left( d \right){J_\varepsilon }\left( \delta \right)} \over {{{\left[ {d,\delta } \right]}^{\sigma + 2\varepsilon + it}}}},} which are connected with usual zeta approximating sums. Some other related summatory functions involving Jordan’s function are considered and estimated in this paper.

Lower Bounds on the Largest Inhomogeneous Approximation Constant

Mon, 24 Feb 2025 00:00:00 GMT

For a given irrational number α and a real number γ in (0, 1) one defines the two-sided inhomogeneous approximation constant M(α,γ):=lim| n |→∞ inf | n |‖ nα-γ ‖, M\left( {\alpha ,\gamma } \right): = \mathop {\lim }\limits_{\left| n \right| \to \infty } \,\,\inf \,\,\,\left| n \right|\left\| {n\alpha - \gamma } \right\|, and the case of worst inhomogeneous approximation for α ρ(α):=supγ∉{ m+lα:m,l∈ℤ }M(α,γ). \rho \left( \alpha \right): = \mathop {\sup }\limits_{\gamma \notin \left\{ {m + l\alpha :m,l \in \mathbb Z} \right\}} M\left( {\alpha ,\gamma } \right). We are interested in lower bounds on ρ(α)in terms of R := lim infi→∞ ai, where the ai are the partial quotients in the negative (i.e., the ‘round-up’) continued fraction expansion of α. We obtain bounds for any R ≥ 3 which are best possible when R is even (and asymptotically precise when R is odd). In particular when R ≥ 3 ρ(α)≥163+8=118.3923…, \rho \left( \alpha \right) \ge {1 \over {6\sqrt 3 + 8}} = {1 \over {18.3923 \ldots }}, and when R ≥ 4, optimally, ρ(α)≥143+2=18.9282…, \rho \left( \alpha \right) \ge {1 \over {4\sqrt 3 + 2}} = {1 \over {8.9282 \ldots }},

Corrigendum to: On Jordan Double Sums and Related Summatory Functions (Uniform Distribution Theory 18 (2023) No. 2, 57–76.)

Mon, 24 Feb 2025 00:00:00 GMT

Vera T. Sós (1930–2023) an Obituary

Mon, 24 Feb 2025 00:00:00 GMT

Sumsets with Prescribed Lower and Upper Asymptotic Densities II

Mon, 24 Feb 2025 00:00:00 GMT

We show that for natural numbers h, k, such that k ≥ 2, and for real numbers α1, α2, such that 0 <α1 ≤ α2 < 1, there exists a subset A of ℕ = {0, 1, 2,... }, such that for all finite subsets B of ℕ such that max B − min B = h, the lower and upper asymptotic densities of kA + B are α1 and α2, respectively.

Uniform Distribution of αn Modulo One for A family of Integer Sequences

Mon, 24 Feb 2025 00:00:00 GMT

We show that the sequence (αn)n∈ℬ is uniformly distributed modulo 1, for every irrational α, provided ℬ belongs to a certain family of integer sequences, which includes the prime, almost prime, squarefree, practical, densely divisible and lexicographical numbers. We also give an estimate for the discrepancy if α has finite irrationality measure.

On the Distribution Modulo one of the a-Points of the k th Derivative of an L-Function in the Selberg Class

Mon, 24 Feb 2025 00:00:00 GMT

Let F be an L-function from the Selberg class, F (k) be the kth derivative of F and a be an arbitrary fixed complex number. The solutions of F (k)(s)= a are called a-points. In this paper, we present a new zone without a-points of F (k) and we show that if F has a polynomial Euler product and satisfies the analogue of the Lindelöf hypothesis, then the a-points of F (k) are uniformly distributed modulo one.