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Thomas Prellberg

Publications and source records attributed to Thomas Prellberg.

At least 19 recordsLinked to original sources
arXiv2026

No-three-in-line sets on the checkerboard grid

The classical no-three-in-line problem asks for the largest number $D(n)$ of points that can be chosen from an $n\times n$ grid with no three collinear; it remains open whether the elementary upper bound $2n$ is always attainable. We study a checkerboard-restricted variant in which all chosen points are monochromatic under the colouring of the grid by the parity of $x+y$. If $D_{\mathrm{mono}}(n)$ denotes the largest number of monochromatic points with no three collinear, the monochromatic diagonals already give $D_{\mathrm{mono}}(n)\le 2n-2$. The main object of the paper is a four-direction linear-programming relaxation on a fixed colour class, using rows, columns and the two diagonal families of slopes $\pm1$. For the ordinary square-grid problem this relaxation gives the trivial bound; on the checkerboard it is substantially tighter, and for $2\le n\le 16$ its floor agrees with the exact single-colour optimum except at four side lengths, where the gap is one. After symmetry reduction the dual relaxation has three one-dimensional reduced forms, according to the parity of $n$ and the chosen colour class. The central construction is an exact continuum dual certificate for the continuum problem associated with the scaled symmetry-reduced odd-fat case: explicit nonnegative functions $A$ and $B$ satisfying the continuum obstacle inequalities, with objective value the middle real root $α\approx1.5768$ of $401α^3-1744α^2+2240α-768=0$. Combined with a continuum-to-discrete sampling theorem of Aujla, Prellberg and Sandhu, this certificate yields the asymptotic upper bound $D_{\mathrm{mono}}(n)\leαn+O(1)$. Finite LP computations are consistent with $α$ as the exact limiting slope of the relaxation, and the exact small-$n$ data suggest, more speculatively, that the true checkerboard optimum tracks the same scale.

math.CO↗
arXiv2026

Moment obstructions and continuum-to-discrete bounds for checkerboard no-three-in-line sets

Fix one colour class in the checkerboard colouring of an $n\times n$ integer grid, and let $M_4(n,\varepsilon)$ be the largest subset having at most two points in every row, column, and diagonal of slopes $\pm1$. We prove the near-saturation bound $M_4(n,\varepsilon)\leq2n-4$ for $n\geq6$. The proof uses first and second moments of the four line families: a hypothetical set of size $2n-3$ produces row, column, and diagonal deficits whose exact moment identities contradict Cauchy--Schwarz. A finite argument handles $n=6$. The same identity extends to arbitrary deficit multisets. It gives $M_4(n,\varepsilon)\leq2n-d$ whenever $d\geq4$ is an integer and $n\geq3d-4$, and an entirely discrete asymptotic estimate \[ M_4(n,\varepsilon)\leq(\sqrt{21}-3)n+8. \] We also prove a general continuum-to-discrete theorem for the associated four-direction fractional packing problem. Applying it to the exact continuum dual certificate constructed in earlier work yields, for both colours, \[ L_{\mathrm{mono}}(n,\varepsilon)\leqαn+O(1), \qquad α\approx1.5768233968738, \] and hence the same upper bound for $M_4$ and for checkerboard no-three-in-line sets.

math.CO↗
arXiv2026

Collision-Generated Compression for Homogeneous Keller Maps

We associate to a collision $p\ne q$ of a homogeneous Keller map $F=\operatorname{id}+h$ the polarization subalgebra generated by $p$ and $q$. This collision hull is the smallest invariant linear subspace containing the collision; the restricted map remains a noninjective Keller map, and its dimension controls that of the standard symmetric lift. The construction is compatible with scalar extension and equivariant under linear conjugacy, so it gives a canonical linear carrier for a marked failure of injectivity. We compute this carrier exactly in two recent cubic-homogeneous reductions of the three-variable Jacobian counterexample. For Thompson's 24-variable map, the growth $2,4,11,20,20$ recovers MacFarlane's 20-dimensional invariant subspace. For the 19-variable homogenization of Van Rijn's 12-variable degree-three map, the growth $2,4,11,19,19$ fills the whole space. Hence no invariant linear restriction retaining the displayed collision can improve the corresponding 38-variable symmetric lift. We give the resulting 340-monomial homogeneous quartic explicitly: it is Hessian-nilpotent, violates Zhao's Vanishing Conjecture, and its gradient Keller map has an exact collision over $\mathbb{Q}(i)$. The analogous 20-variable application gives the previously studied 40-variable, 350-monomial quartic. All finite calculations are checked by exact companion code.

math.AG↗
arXiv2026

Cayley-graph density of Thompson's group F: local deletions and finite-window bounds

Let $Γ$ be the Cayley graph of Thompson's group $F$ with its standard generators. Belk and Brown constructed finite marked-forest subgraphs of limiting density $7/2$, and Guba showed that deleting certain low-degree vertices gives density greater than $3.5004$. We refine this approach in two directions. First, a deterministic interval-deletion rule followed by root-sensitive triple and pair deletions yields finite induced subgraphs with \[ \operatorname{dens}(Γ)>3.50074529. \] The interval rule is evaluated by a nine-state recurrence, while the simultaneous deletion conditions created by split and merge operations are computed exactly by a common-suffix first-passage argument. Second, we determine the joint distribution of the categories of a tree and its two children and reduce optimization over every fixed category-window retention rule to a weighted densest-subgraph problem. An explicit edge-allocation certificate shows that, for every fixed $a\ge0$, the optimal limiting density of a rule depending on the window $[-a,1]$ is exactly $7/2$. Thus arbitrary finite left context together with one right-hand category does not improve the Belk--Brown limit, whereas the root-sensitive whole-segment construction does. All constants are explicit elements of $\mathbb Q(\sqrt3,\sqrt{2\sqrt3-1})$.

math.CO↗
arXiv2026

Three Irrationality Results for the Dilogarithm: Rhin--Viola and Viola--Zudilin Constructions at $-1/4$, $1/5$, and $-1/3$

We prove the irrationality of \[ \Li_2(-1/4),\qquad \Li_2(1/5),\qquad \Li_2(-1/3). \] The three arguments form a natural progression. The first is an endpoint completion of the five-parameter Rhin--Viola method. Rhin and Viola's 2019 treatment already supplies the negative-argument continuation, permutation invariance and factorial divisor; at $z=-4$ we use three bounded shifts, an elementary fixed-contour estimate and a four-term recurrence to remove the remaining nonvanishing problem in the complex-saddle regime. For $1/5$ we pass to the six-parameter Viola--Zudilin family and apply the Rhin--Viola factorial transformations term by term inside two binomial expansions of the same integral. This inherited divisor crosses the arithmetic threshold that the unrefined five-parameter construction does not reach in our computations. The same crossbreed, combined with the negative-argument continuation, proves the result at $-1/3$. All proof-critical finite inequalities are accompanied by exact executable certificates. Supplementary computations propagate $10{,}000$ exact primitive coefficient pairs for each of $1/5$ and $-1/3$, with complete gcd removal and no failure of the certified smallness or adjacent nonproportionality checks.

math.NT↗
arXiv2026

Area and water-capacity statistics for upper hulls of Dyck paths

We study Dyck paths refined simultaneously by proper area and water capacity, where water capacity is measured above the path and below its lattice-path upper hull. The finite-height ingredients used in the enumeration are classical bounded-height area-polynomial and continued-fraction objects. The upper-hull decomposition produces a coupled area--capacity substitution, which gives an exact four-variable height expansion with denominator branches indexed by the height levels. The full generating function is asymmetric in the two weights, while the height summands admit a symmetric unreduced denominator representation under interchange of the area and capacity weights. In the open square $0<p,q<1$, we prove that the length radius of $G(x,1,p,q)$ is the minimum of the positive real denominator branches. The proof combines uniform normal convergence of the height expansion below this first branch with a Perron-root representation of the branch locations and an interval log-submodularity theorem for spectral radii of weighted paths. On the diagonal $p=q=s$, the classical Chebyshev specialisation gives explicit branch crossings and a $(1-s)^{2/3}$ branch-envelope accumulation law at the Dyck critical point.

math.CO↗
arXiv2025

Three methods, one problem: Classical and AI approaches to no-three-in-line

The No-Three-In-Line problem asks for the maximum number of points that can be placed on an n by n grid with no three collinear, representing a famous problem in combinatorial geometry. While classical methods like Integer Linear Programming (ILP) guarantee optimal solutions, they face exponential scaling with grid size, and recent advances in machine learning offer promising alternatives for pattern-based approximation. This paper presents the first systematic comparison of classical optimization and AI approaches to this problem, evaluating their performance against traditional algorithms. We apply PatternBoost transformer learning and reinforcement learning (PPO) to this problem for the first time, comparing them against ILP. ILP achieves provably optimal solutions up to 19 by 19 grids, while PatternBoost matches optimal performance up to 14 by 14 grids with 96% test loss reduction. PPO achieves perfect solutions on 10 by 10 grids but fails at 11 by 11 grids, where constraint violations prevent valid configurations. These results demonstrate that classical optimization remains essential for exact solutions while AI methods offer competitive performance on smaller instances, with hybrid approaches presenting the most promising direction for scaling to larger problem sizes.

cs.AI↗
arXiv2024

Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series

We investigate radial statistics of zeros of hyperbolic Gaussian Analytic Functions (GAF) of the form $φ(z) = \sum_{k\ge 0} c_k z^k$ given that $|φ(0)|^2=t$ and assuming coefficients $c_k$ to be independent standard complex normals. We obtain the full conditional distribution of $N_q$, the number of zeros of $φ(z)$ within a disk of radius $\sqrt{q}$ centred at the origin, and prove its asymptotic normality in the limit when $q\to 1^{-}$, the limit that captures the entire zero set of $φ(z)$. In the same limit we also develop precise estimates for conditional probabilities of moderate to large deviations from normality. Finally, we determine the asymptotic form of $P_k(t;q)=\mathrm{Prob} \{ N_q= k | |φ(0)|^2=t \}$ in the limit when $k$ is kept fixed whilst $q$ approaches 1. To leading order, the hole probability $P_0(t;q)$ does not depend on $t$ for $t>0$ but yet is different from that of $P_0(t=0;q)$ and coincides with the hole probability for unconditioned hyperbolic GAF of the form $\sum_{k\ge 0} \sqrt{k+1}\, c_k z^k$. We also find that asymptotically as $q \to 1^{-}$, $P_k(t;q)= e^t P_{k}(0;q)$ for every fixed $k \ge 1$ with $P_{k}(0;q)= \mathrm{Prob} \{ N_q =k-1 \}$.

math.PR↗
arXiv2024

Improving Convergence of Generalised Rosenbluth Sampling for Branched Polymer Models by Uniform Sampling

Sampling with the Generalised Atmospheric Rosenbluth Method (GARM) is a technique for estimating the distributions of lattice polymer models that has had some success in the study of linear polymers and lattice polygons. In this paper we will explain how and why such sampling appears not to be effective for many models of branched polymers. Analysing the algorithm on a simple binary tree, we argue that the fundamental issue is an inherent bias towards extreme configurations that is costly to correct with reweighting techniques. We provide a solution to this by applying uniform sampling methods to the atmospheres that are central to GARM. We caution that the ensuing computational complexity often outweighs the improvements gained.

cond-mat.stat-mech↗
arXiv2023

On the universality class of the special adsorption point of two-dimensional lattice polymers

In recent work [PRE 100, 022121 (2019)] evidence was found that the surface adsorption transition of interacting self-avoiding trails (ISATs) placed on the square lattice displays a non-universal behavior at the special adsorption point (SAP) where the collapsing polymers adsorb. In fact, different surface exponents $ϕ^{(s)}$ and $1/δ^{(s)}$ were found at the SAP depending on whether the surface orientation is horizontal (HS) or diagonal (DS). Here, we revisit these systems and study other ones, through extensive Monte Carlo simulations, considering much longer trails than previous works. Importantly, we demonstrate that the different exponents observed in the reference above are due to the presence of a previously unseen surface-attached-globule (SAG) phase in the DS system, which changes the multicritical nature of the SAP and is absent in the HS case. By considering a modified horizontal surface (mHS) where the trails are forbidden of having two consecutive steps along it, resembling the DS situation, a stable SAG phase is found in the phase diagram, and both DS and mHS systems present similar $1/δ^{(s)}$ exponents at the SAP, being $1/δ^{(s)} \approx 0.44$, whilst $1/δ^{(s)} \approx 0.34$ in the HS case. Intriguingly, while $ϕ^{(s)} \approx 1/δ^{(s)}$ is found for the DS and HS scenarios, as expected, in the mHS case $ϕ^{(s)}$ is about $10$\% smaller than $1/δ^{(s)}$. These results strongly indicate that at least two universality classes exist for the SAPs of adsorbing ISATs on the square lattice.

cond-mat.soft↗
arXiv2020

Skew Schur Function Representation of Directed Paths in a Slit

In this work, we establish a general relationship between the enumeration of weighted directed paths and skew Schur functions, extending work by Bousquet-Mélou, who expressed generating functions of discrete excursions in terms of rectangular Schur functions.

math.CO↗
arXiv2019

Adsorption of 2d polymers with two- and three-body self-interactions

Using extensive Monte Carlo simulations, we investigate the surface adsorption of self-avoiding trails on the triangular lattice with two- and three-body on-site monomer-monomer interactions. In the parameter space of two-body, three-body, and surface interaction strengths, the phase diagram displays four phases: swollen (coil), globule, crystal, and adsorbed. For small values of the surface interaction, we confirm the presence of swollen, globule, and crystal bulk phases. For sufficiently large values of the surface interaction, the system is in an adsorbed state, and the adsorption transition can be continuous or discontinuous, depending on the bulk phase. As such, the phase diagram contains a rich phase structure with transition surfaces that meet in multicritical lines joining in a single special multicritical point. The adsorbed phase displays two distinct regions with different characteristics, dominated by either single or double layer adsorbed ground states. Interestingly, we find that there is no finite-temperature phase transition between these two regions though rather a smooth crossover.

cond-mat.stat-mech↗
arXiv2019

Enumerating path diagrams in connection with $q$-tangent and $q$-secant numbers

We enumerate height-restricted path diagrams associated with $q$-tangent and $q$-secant numbers by considering convergents of continued fractions, leading to expressions involving basic hypergeometric functions. Our work generalises some results by M. Josuat-Vergés for unrestricted path diagrams [European Journal of Combinatorics 31 (2010) 1892].

math.CO↗
arXiv2018

Exact solution of pulled, directed vesicles with sticky walls in two dimensions

We analyse a directed lattice vesicle model incorporating both the binding-unbinding transition and the vesicle inflation-deflation transition. From the exact solution we derive the phase diagram for this model and elucidate scaling properties around the binding-unbinding critical point in this larger parameter space. We also consider how the phase diagram changes when a perpendicular force is applied to the end of a directed vesicle.

math-ph↗
arXiv2018

Multicritical Scaling in a Lattice Model of Vesicles

Vesicles, or closed fluctuating membranes, have been modeled in two dimensions by self-avoiding polygons, weighted with respect to their perimeter and enclosed area, with the simplest model given by area-weighted excursions. These models generically show a tricritical phase transition between an inflated and a crumpled phase, with a scaling function given by the logarithmic derivative of the Airy function. Extending such a model, we find realizations of multicritical points of arbitrary order, with the associated multivariate scaling functions expressible in terms of generalized Airy integrals, as previously conjectured by John Cardy. This work therefore adds to the small list of models with a critical phase transition, for which exponents and the associated scaling functions are explicitly known.

cond-mat.stat-mech↗
arXiv2017

Solution of semi-flexible self-avoiding trails on a Husimi lattice built with squares

We study a model of semi-flexible self-avoiding trails, where the lattice paths are constrained to visit each lattice edge at most once, with configurations weighted by the number of collisions, crossings and bends, on a Husimi lattice built with squares. We find a rich phase diagram with five phases: a non-polymerised phase (${\bf NP}$), low density (${\bf P1}$) and high density (${\bf P2}$) polymerised phases, and, for sufficiently large stiffness, two additional anisotropic (nematic) (${\bf AN1}$ and ${\bf AN2}$) polymerised phases within the ${\bf P1}$ phase. Moreover, the {\bf AN1} phase which shows a broken symmetry with a preferential direction, is separated from the ${\bf P1}$ phase by the other nematic ${\bf AN2}$ phase. Although this scenario is similar to what was found in our previous calculation on the Bethe lattice, where the ${\bf AN-P1}$ transition was discontinuous and critical, the presence of the additional nematic phase between them introduces a qualitative difference. Other details of the phase diagram are that a line of tri-critical points may separate the ${\bf P1}-{\bf P2}$ transition surface into a continuous and a discontinuous portion, and that the same may happen at the ${\bf NP}-{\bf P1}$ transition surface, details of which depend on whether crossings are allowed or forbidden. A critical end-point line is also found in the phase diagram.

cond-mat.stat-mech↗
arXiv2017

Area-width scaling in generalised Motzkin paths

We consider a generalised version of Motzkin paths, where horizontal steps have length $\ell$, with $\ell$ being a fixed positive integer. We first give the general functional equation for the area-length generating function of this model. Using a heuristic ansatz, we derive the area-length scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schröder paths, before generalising our approach to arbitrary $\ell$. We then derive an expression for the generating function of Schröder paths and analyse the scaling behaviour of this function rigorously in the vicinity of the tri-critical point of the model by applying the method of steepest descents for the case of two coalescing saddle points. Our results show that for Dyck and Schröder paths, the heuristic scaling ansatz reproduces the rigorous results.

cond-mat.stat-mech↗