arXiv subjects

John M. Campbell

Publications and source records attributed to John M. Campbell.

At least 19 recordsLinked to original sources

arXiv2026

Pinch's conjecture on $a$-convexity

For $a$ in $\mathbb{R}$, a subset $V$ contained in $\mathbb{R}^{n}$ is said to be $a$-convex if $x, y \in V \Longrightarrow a x + (1-a) y \in V$. According to Pinch [Math. Proc. Cambridge Philos. Soc., 1985], the $a$-convex hull of $V$ is the intersection of all of the $a$-convex subsets of $\mathbb{R}^{n}$ that contain $V$, and Pinch also defines $D(a)$ as the $a$-convex hull of $\{ 0, 1 \}$ in $\mathbb{R}^{1}$. Pinch conjectured that if $a$ is a totally real algebraic integer and $D(a)$ has no limit points, then every algebraic conjugate of $a$ other than $a$ is in $(0, 1)$. We succeed in proving this conjecture, which seems to have remained open.

math.NT↗

arXiv2026

On a conjecture concerning antipodal labelings for cycles

Let $G$ denote a finite, connected, simple graph, and let $D$ denote its diameter, and let $d_{G}(u, v)$ denote the distance between vertices $u$ and $v$ in $V(G)$. An antipodal labeling of $G$ is a mapping $f\colon V(G) \to \mathbb{N}_{0}$ such that, for each pair $(u, v)$ consisting of distinct vertices in $V(G)$, the relation $ |f(u) - f(v)| \geq D - d_{G}(u, v) $ holds. The span of $f$ is then defined so that $\operatorname{sp}(f) = \max\{ f(u) - f(v) : u, v \in V(G) \}$. The antipodal number of $G$, denoted with $\operatorname{an}(G)$, may then be defined as the minimum possible span among all antipodal labelings of $G$. Juan and Liu [Ars Combin., 2012] proved the upper bound $\operatorname{an}(C_{4k}) \leq 2k^2 - 1$, and conjectured that $\operatorname{an}(C_{4k}) = 2k^2 - 1$ for each positive integer $k$, and Juan and Liu verified this conjecture for $k \leq 5$. We succeed in proving Juan and Liu's conjecture, which seems to have remained open. The openness of Juan and Liu's conjecture has been noted, over the years, by many authors, including Rao et al. [Contrib. Discrete Math., 2015] and Saha et al. [Theory Comput. Syst., 2022].

math.CO↗

arXiv2026

Abelian-normal decimal expansions

A landmark paper on normal numbers due to Champernowne [J. London Math. Soc., 1933] provided the first explicit construction of a normal number. Since many research works have concerned normality-preserving selection rules and operations on the sequence of digits of a given normal number, this leads us to introduce the concept of an abelian-normal number, in such a way so that subwords that are equivalent up to permutations are counted as being equivalent, and so that the number of occurrences of equivalent subwords appearing, up to a given point in a decimal expansion, is then normalized by the size of the associated equivalence class. This may be thought of as giving rise to an interdisciplinary area linking combinatorics on words and probabilistic number theory. Since every normal number is abelian-normal, this raises the question as to whether or not there exists an abelian-normal number that is not normal. We succeed in constructing such a constant in an explicit way.

math.CO↗

arXiv2026

Double-real corrections to color singlet decay in a parton-shower inspired scheme

We introduce a local infrared subtraction method for next-to-next-to-leading order QCD calculations in color singlet decays, which is modeled on the evolution algorithm of a parton shower. Overlapping singularities in the multipole radiation pattern are disentangled by partial fractioning, and the kinematics mapping corresponds to iterated next-to-leading order kinematics. We verify that the double-real remainder to $e^+e^-\to\;q\bar{q}$ is rendered finite in the single and double unresolved limits and investigate the numerical convergence of the Monte-Carlo integral. We compute the phase-space integrals of the scalar counterterms in the back-to-back configuration, both analytically and with the help of numerical techniques based on sector decomposition.

hep-ph↗

arXiv2026

On two conjectures on triangulations of 2-manifolds

For a closed, connected 2-manifold $M$, and for a triangulation $T$ of $M$, we write $V(T)$ in place of the vertex set associated with $T$. A cyclic coloration of a triangulation $T$ of $M$ refers to a face coloring of $T$ such that: For each $v \in V(T)$, the faces incident to $v$ have distinct colors. Chen and Lawrencenko [Yokohama Math. J., 1999] conjectured that there exists a constant $C(M)$ (depending only on $M$) such that $|V(T)| + C(M)$ colors suffice for there to exist a cyclic coloration of a triangulation $T$ of $M$. We prove this conjecture, using a greedy algorithm related to the Euler-Poincaré formula for surface triangulations. Chen and Lawrencenko also conjectured that: If $M$ is not the projective plane and $T$ is a triangulation of $M$ that is minimal with respect to the number of vertices, then $ξ(T) = |V(T)| = V_{\min}(M)$, where $V_{\min}(M)$ denotes the minimum possible number of vertices among all triangulations of the 2-manifold $M$, and where $ξ(T)$ denotes the minimal value $k$ such that $T$ admits a cyclic coloration with $k$ colors. We disprove this latter conjecture via an explicit counterexample, using an 8-vertex triangulation of the Klein bottle with 16 faces. It appears that both of the Chen-Lawrencenko conjectures have remained open, prior to our work.

math.GT↗

arXiv2026

The finitude of the fibers of the complementary Bell numbers

Subbarao and Verma introduced, in 1999, a number of open problems concerning the sequence $(f(n))_{n \geq 0}$ of complementary Bell numbers, which may be defined via Bell polynomials $B_{n}(x) = \sum_{k=0}^{n} \left\{ \begin{smallmatrix} n \\ k \end{smallmatrix} \right\} x^k$ so that $f(n) = B_{n}(-1)$. Yang [Electron. J. Combin., 2001] subsequently solved the first two problems from Subbarao and Verma, but the third such problem has remained open, to the best of our knowledge. The first part of this third problem asks whether or not $f(n)$ takes any given value only a finite number of times. We solve this problem in the affirmative, through a combined application of finite difference-based methods, partial Motzkin paths, the completeness of the Tate algebra with respect to the Gauss norm, and Strassmann's theorem.

math.CO↗

arXiv2026

Disproofs of two conjectures concerning nondeficient numbers

A positive integer $n$ is said to be nondeficient if $σ(n) \geq 2n$. Letting the positive divisors of a positive integer $n$ be written as $1 = d_0 < d_1 < \cdots < d_k < d_{k+1} = n$, and letting $\mathcal{S}$ denote a set of integers, if there exist values $λ_j \in \mathcal{S}$ such that $1 + \sum_{j=1}^{k} λ_j d_j = n$, then $n$ is said to be an $\mathcal{S}$-perfect number. Ross, in 2024, introduced the study of $\mathcal{S}$-perfect numbers, and concluded with two conjectures that each concern both $\{ -1, 1 \}$-perfect numbers and nondeficient numbers. We disprove both of these conjectures.

math.NT↗

arXiv2026

On a conjecture on $k$-antichains in the unit $n$-cube

Let $[0, 1]^{n} \subseteq \mathbb{R}^{n}$ be endowed with its pointwise order, and let $k$ be a positive integer. A subset $A$ of $[0, 1]^{n}$ is said to be a \emph{$k$-antichain} if $\operatorname{card}(A \cap C) \leq k$ for each chain $C \subseteq [0, 1]^{n}$. Letting $\mathcal{H}^{m}$ denote the $m$-dimensional Hausdorff outer measure, Pelekis and Vlasák [Publ.\ Math.\ Debrecen, 2020] conjectured that there exists a $k$-antichain $A \subseteq [0, 1]^{n}$ satisfying $\mathcal{H}^{n-1}(A) = k n$, and proved the special case of this conjecture for $n = 2$, whereas Janzer [Mathematika, 2020] proved the $k = 1$ case of Pelekis and Vlasák's conjecture. This conjecture is motivated by a result due to Erdős on $k$-antichains in $\{ 0, 1 \}^{n}$. We prove Pelekis and Vlasák's conjecture in full generality, thus establishing that their upper bound $\mathcal{H}^{n-1}(A) \leq k n$ is sharp for $k$-antichains $A$ in $[0, 1]^{n}$.

math.CO↗

arXiv2026

On a problem on a generalization of Euler's totient function

Büyükaşik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $φ(n)$, by setting $φ_k(n) = \sum_{a} a^k$ for $a \in [1, n]$ such that $(a, n) = 1$, with $φ_0(n) = φ(n)$. Letting $\mathcal{D}_{s} = \{ k \geq s : \forall n \geq 1 \ φ_s(n) \mid φ_k(n) \}$, Büyükaşik et al. proved that $\mathcal{D}_{s}$ is finite for each $s \geq 0$, and conjectured that $\mathcal{D}_{1} = \{ 1, 3, 15 \}$ and provided computations to support this conjecture. We succeed in proving this conjecture, using an argument based on our extensive interactions with GPT-5.5 Pro.

math.NT↗

arXiv2026

Meta-automatic Sequences

Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's Q-sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the $Q$-sequence is $2$-automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We exhibit two binary meta-automatic sequences $M_1$ and $M_2$ whose defining recurrences do not satisfy the Allouche-Shallit automaticity criterion directly, and this is formalized in our paper. For each of these integer sequences $M_1$ and $M_2$, we prove explicit DFAO evaluations, together with $4$-uniform morphisms, and we also consider the factor complexities of these sequences.

math.NT↗

arXiv2026

On the binary digits of the Erdős-Borwein constant

In a landmark paper on arithmetical properties of Lambert series, Erdős proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erdős-Borwein constant. Crandall, in 2012, studied properties of the base-2 expansion of this constant, and left the following as an open problem: Does the string $11$ occur infinitely often in the base-2 expansion of $E$? This open problem was also subsequently noted by Shallit. We succeed in introducing a full proof that solves Crandall's problem in the affirmative. Our proof combines a congruence construction in the spirit of Erdős and an estimate due to Alford, Granville, and Pomerance for the counting function for primes in arithmetic progressions. Our argument was developed through extensive interactions with GPT-5.5 Pro.

math.NT↗

arXiv2026

On divisor sums due to Erdős and Ramanujan

Let $d(n)$ denote the number of divisors of a positive integer $n$. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum $\sum_{n \leq x} \frac{1}{d(n)}$, with Ramanujan having introduced an asymptotic formula for this sum with an explicit evaluation for the constant $A_1$ for the leading term $A_1 \frac{x}{\sqrt{\log x}}$. Gabdullin et al. recently considered a hybrid of this problem and the Titchmarsh divisor problem concerning $\sum_{p\leq x} d(p-1)$, proving that $$\sum_{p\leq x} \frac{1}{d(p-1)} \asymp \frac{x}{(\log x)^{3/2}}.$$ This result, together with Erdős's asymptotic formula $\sum_{n \leq x} d(d(n)) \sim c \, x \log \log x $ for a constant $c \in (0, \infty)$, lead us to consider the hybrid $\sum_{n \leq x} \frac{1}{d(d(n))}$ of the Erdős and Ramanujan divisor sums. The presence of the reciprocal significantly complicates the analysis, as it amplifies the contribution of integers for which $d(d(n))$ is exceptionally small. In this paper, we prove that $$\sum_{n \leq x} \frac{1}{d(d(n))} \asymp \frac{x}{ \log \log x}, $$ through a combined application of Golomb's estimate for powerful numbers and Turán's quantitative form of the Hardy-Ramanujan theorem.

math.NT↗

arXiv2026

Subword enumeration up to stack-sorting equivalence

Defant and Kravitz introduced generalizations of West's stack-sorting map $s$ from permutations to finite words. This raises questions as to how such generalizations could be applied in the field of combinatorics on words. The Defant-Kravitz generalizations of $s$ depend on how repeated occurrences of the same character within a word may be repositioned, according to their $\textsf{tortoise}$ and $\textsf{hare}$ operations. As demonstrated in this paper, these operations provide a natural way of extending abelian complexity functions for infinite sequences, in a way that gives light to structural properties associated with infinite words. We apply these new ideas to two famous infinite words: the paperfolding word and the Thue-Morse word. In the case of the Thue-Morse word, we discover an interesting connection to the previous work of several authors, such as de Luca and Varricchio, on the ``special'' factors of the Thue-Morse word. This may be seen as providing a basis for a new and interdisciplinary area linking the combinatorics about the stack-sorting of permutations with the field of combinatorics on words.

math.CO↗

arXiv2026

Kronecker coefficients via the Giambelli identity for Schur functions

One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{λ\, μ\, ν}$. A notable advance in this direction is due to Blasiak, who proved a combinatorial interpretation in terms of colored Yamanouchi tableaux for the case whereby one of the indexing partitions is hook-shaped. In this paper, we introduce a framework for the evaluation and combinatorial interpretation of Kronecker coefficients, combining a Schur function identity of Littlewood, the Giambelli identity for Schur functions, and Blasiak's combinatorial rule. This framework reduces the study of Kronecker coefficients to alternating sums involving hook-indexed cases. As an application of this framework, we obtain combinatorial interpretations of $g_{t, h^{(1)}, h^{(2)}}$ for two-row partitions $t$ and hook-like partitions $h^{(1)}$ and $h^{(2)}$ satisfying natural conditions. More broadly, our approach provides a systematic method for extending hook-based combinatorial rules to wider families of Kronecker coefficients.

math.CO↗

arXiv2026

A further q-analogue of Gosper's strange series

Recently, the second author [Ramanujan J. 2026] introduced and proved a $q$-series identity that appears to provide the first known $q$-analogue of an evaluation for a ${}_{2}F_{1}$-series known as \emph{Gosper's strange series}. Yamaguchi's derivation of this $q$-analogue relies on three-term relations for ${}_{2}ϕ_{1}$-series along with Heine's transformation of ${}_{2}ϕ_{1}$-series. In this note, we introduce and prove, using a $q$-analogue of a series evaluation technique relying on an Abel-type summation lemma, a further $q$-analogue of Gosper's ${}_{2}F_{1}$-identity that is inequivalent to Yamaguchi's $q$-analogue, and we also apply this technique to construct an alternative and simplified proof of Yamaguchi's $q$-analogue, together with a ${}_{3}ϕ_{2}$-series variant of Heine's $q$-analogue of Gauss's hypergeometric formula, a ${}_{6}ϕ_{5}$-series variant and two ${}_{4}ϕ_{3}$-series variants of the $q$-analogue of Kummer's identity due to Bailey and Daum, along with a $q$-analogue of a result obtained by Cantarini [Ramanujan J. 2022] via Fourier-Legendre theory and related to Ramanujan's first series for $\frac{1}π$.

math.CA↗

arXiv2026

The construction of $q$-analogues via $_3ϕ_2$-series and $q$-difference equations

We apply the EKHAD-normalization method given in our recent work to obtain, via the $q$-version of Zeilberger's algorithm, $q$-WZ pairs $(F, G)$ such that $\sum_{k = 0}^{\infty} F(0, k)$ may be expressed as a basic hypergeometric series of the form ${}_{3}ϕ_2$ with multiple free parameters, and in such a way so that $\sum_{k=0}^{\infty} F(0, k) = \sum_{n=0}^{\infty} G(n, 0)$. In contrast to how previous applications of EKHAD-normalization relied on $q$-analogues for specific WZ pairs introduced by Guillera, our multiparameter approach provides a broad framework in the construction of $q$-analogues for accelerated series for universal constants such as $π$. We apply this multiparameter version of EKHAD-normalization to obtain and prove new $q$-analogues for accelerated hypergeometric series attributed to many authors, including (alphabetically) Adamchik and Wagon, Apéry, Chu, Chu and Zhang, Fabry, Guillera, Ramanujan, and Zeilberger.

math.CO↗

arXiv2026

Quadratic irrational analogues of Ramanujan's series for $1/π$

About 40 years ago Jonathan and Peter Borwein discovered the series identity $$ \sum_{n=0}^\infty \frac{(-1)^n(6n)!}{(3n)!(n!)^3} \frac{(A+nB)}{C^{n+1/2}} = \frac{1}{12π} $$ where \begin{align*} A&=1657145277365+212175710912\sqrt{61},\cr B&=107578229802750+13773980892672\sqrt{61},\cr C&=\left(5280(236674+30303\sqrt{61})\right)^3 \end{align*} which adds roughly 25 digits of accuracy per term. They noted that if each of the quadratic irrationals $A$, $B$ and $C$ is replaced by their conjugates, that is, each number $a+b\sqrt{61}$ is changed to $a-b\sqrt{61}$, then the resulting series also converges to a rational multiple of $1/π$. They gave several other examples of quadratic irrational series for $1/π$, and noted that the conjugate series converges to another rational multiple of $1/π$ or in some cases the conjugate series diverges. The purpose of this work is to provide an explanation and classification of such series. Our classification includes Ramanujan's 17 original series, as well as series of the Borweins, Chudnovskys, Sato and others. We extend the classification to genus-zero subgroups $Γ_0(\ell)+$, that is, for each $\ell \in \big\{1,2,3,\ldots,36,38,39,41,42,44,45,46,47,49,50,51,54,55,56,59,60,62,66,69,70, 71,78,87,92,94,95,105,110,119\big\}$ we calculate the Hauptmoduls, associated weight two modular forms, and the corresponding rational and real quadratic irrational series for $1/π$. The classification reveals many interrelations among the different series. For example, we show that the Borweins' series above, and its conjugate, are equivalent by hypergeometric transformation formulas to the level~7 rational series $$ \sum_{n=0}^\infty \left\{\sum_{j=0}^n {n \choose j}^2{2j \choose n} {n+j \choose j}\right\} (11895n+1286) \frac{(-1)^n}{22^{3n+3}} = \frac{1}{π\sqrt{7}}. $$

math.NT↗

arXiv2026

QCD splitting functions beyond kinematical limits

We present a systematic decomposition of QCD splitting functions into scalar dipole radiators and pure splitting remainders up to second order in the strong coupling. The individual components contain terms that are formally sub-leading in soft or collinear scaling parameters, but well understood and universal due to their origin in scalar QCD. The multipole radiator functions which we derive share essential features of the known double-soft and one-loop soft gluon currents, and are not based on kinematical approximations.

hep-ph↗