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Robbert Fokkink

Publications and source records attributed to Robbert Fokkink.

At least 19 recordsLinked to original sources
arXiv2025

Cobham's theorem for the Gaussian integers

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $\alpha=-m+i $- and $\beta=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $\alpha$ and $\beta$ are multiplicatively independent Gaussian integers, and at least one of $\alpha$, $\beta$ is not an $n$-th root of an integer, then any $\alpha$- and $\beta$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $\alpha$-automatic for any root of an integer $\alpha$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations.

math.NT↗
arXiv2025

On Cloitre's hiccup sequences

In 2003, Benoit Cloitre entered a family of sequences in the OEIS that we call hiccup sequences. We collect the various claims, observations, and proofs of properties of these sequences that have been entered in the OEIS over the years, and present a unified approach, inspired by a remarkable theorem of Bosma, Dekking, and Steiner.

math.CO↗
arXiv2025

Anti-recurrence sequences

We extend previous work on anti-recurrence sequences of Kimberling and Moses, Zaslavsky, and Bosma et al. Kimberling and Moses have formulated several questions on these sequences, which can be combined into the meta-conjecture that anti-recurrence sequences are sums of linear progressions and automatic sequences. We solve this conjecture under a restriction on the linear form that generates the anti-recurrence.

math.NT↗
arXiv2025

On the records and zeros of a deterministic random walk

We settle two questions on sequence A120243 in the OEIS that were raised by Clark Kimberling and partly solve a conjecture of Van de Lune and Arias de Reyna. We extend Kimberling's questions to the framework of deterministic random walks, automatic sequences, and linear recurrences. Our results indicate that there may be a deeper connection between these structures. In particular, we conjecture that the records of deterministic random walks are $\xi$-Ostrowski automatic for a quadratic rotation number $\xi$.

math.DS↗
arXiv2025

Using Walnut to solve problems from the OEIS

We use the automatic theorem prover Walnut to resolve various open problems from the OEIS and beyond. Specifically, we clarify the structure of sequence A260311, which concerns runs of sums of upper Wythoff numbers. We extend a result of Hajdu, Tijdeman, and Varga on polynomials with nonzero coefficients modulo a prime. Additionally, we settle open problems related to the anti-recurrence sequences A265389 and A299409, as well as the subsumfree sequences A026471 and A026475. Our findings also give rise to new open problems.

math.NT↗
arXiv2023

Multiplayer boycotts in convex games

We extend the notion of boycotts in cooperative games from one-on-one boycotts between single players to boycotts between coalitions. We prove that convex games offer a proper setting for studying the impact of boycotts. Boycotts have a heterogeneous effect. Individual players that are targeted by many-on-one boycotts suffer most, while non-participating players may actually benefit from a boycott.

cs.GT↗
arXiv2023

Search and Rescue on a Poset

A Search and Rescue game (SR game) is a new type of game on a graph that has quickly found applications in scheduling, object detection, and adaptive search. In this paper, we broaden the definition of SR games by putting them into the context of ordered sets and Bayesian networks, extending known solutions of these games and opening up the way to further applications.

cs.GT↗
arXiv2023

The Pell Tower and Ostronometry

Conway and Ryba considered a table of bi-infinite Fibonacci sequences and discovered new interesting patterns. We extend their considerations to tables that are defined by the recurrence $X_{n+1}=dX_n+X_{n-1}$ for natural numbers $d$. In our search for new patterns we run into a Red Wall and exotic numeration systems.

math.CO↗
arXiv2022

Some remarks on the Game of Cycles

The Game of Cycles is an impartial game on a planar graph that was introduced by Francis Su. In this short note we address some questions that have been raised on the game, and raise some further questions.

math.CO↗
arXiv2022

Corner the Empress

Wythoff Nim aka Corner the Lady is a classic combinatorial game. A Queen is placed on an infinite chess board and two players take alternate turns, moving the Queen closer to the corner. The first player that corners the Queen wins. What happens if the Queen gets superior powers and is able to go off diagonal? In this paper we study the intriguing patterns that emerge from such games.

math.CO↗
arXiv2022

Automorphism groups of random substitution subshifts

We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.

math.DS↗
arXiv2020

Some inequalities on Binomial and Poisson probabilities

Let $S$ and $X$ be independent random variables, assuming values in the set of non-negative integers, and suppose further that both $\mathbb{E}(S)$ and $\mathbb{E}(X)$ are integers satisfying $\mathbb{E}(S)\ge \mathbb{E}(X)$. We establish a sufficient condition for the tail probability $\mathbb{P}(S\ge \mathbb{E}(S))$ to be larger than $\mathbb{P}(S+X\ge \mathbb{E}(S+X))$. We also apply this result to sums of independent binomial and Poisson random variables.

math.PR↗
arXiv2020

"Stay Nearby or Get Checked": A Covid-19 Lockdown Exit Strategy

This paper repurposes the classic insight from network theory that long-distance connections drive disease propagation into a strategy for controlling a second wave of Covid-19. We simulate a scenario in which a lockdown is first imposed on a population and then partly lifted while long-range transmission is kept at a minimum. Simulated spreading patterns resemble contemporary distributions of Covid-19 across nations, regions, and provinces, providing some model validation. Results suggest that the proposed strategy may significantly flatten a second wave. We also find that post-lockdown flare-ups remain local longer, aiding geographical containment. Public policy may target long ties by heavily focusing medical testing and mobility tracking efforts on traffic and transport. This policy can be communicated to the general public as a simple and reasonable principle: Stay nearby or get checked.

cs.SI↗
arXiv2020

Optimizing stakes in simultaneous bets

We want to find the convex combination $S$ of iid Bernoulli random variables that maximizes $\textbf{P}(S\geq t)$ for a given threshold~$t$. Cs\'oka conjectured that such an $S$ is an average if $t\geq p$, where $p$ is the success probability of the Bernoulli random variables. We prove this conjecture for a range of $p$ and $t$.

math.PR↗
arXiv2019

A modification of Wythoff's Nim

We modify Wythoff's game by allowing an additional move, which we call a "split", and show how the $P$-positions are coded by the Tribonacci word. We analyze the table of letter positions of arbitrary $k$-bonacci words and find a $\mathrm{mex}$-rule that generates the Quadribonacci table.

math.CO↗
arXiv2018

On Submodular Search and Machine Scheduling

Suppose some objects are hidden in a finite set $S$ of hiding places which must be examined one-by-one. The cost of searching subsets of $S$ is given by a submodular function and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of $S$ that finds all the objects in minimal expected cost. This problem is NP-hard and we give an efficient combinatorial $2$-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application $1|prec|\sum w_A h(C_A)$, where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function $h$ of the completion times of {\em subsets} of jobs. We go on to give better approximations for submodular functions with low {\em total curvature} and we give a full solution when the problem is what we call {\em series-parallel decomposable}. Next, we consider a zero-sum game between a cost-maximizing Hider and a cost-minimizing Searcher. We prove that the equilibrium mixed strategies for the Hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the series-parallel decomposable case, giving approximately optimal strategies in other cases.

math.OC↗
arXiv2016

On automatic subsets of the Gaussian integers

Suppose that $a$ and $b$ are multiplicatively independent Gaussian integers, that are both of modulus~$\geq \sqrt 5$. We prove that there exist a $X\subset \mathbb Z[i]$ which is $a$-automatic but not $b$-automatic. This settles a problem of Allouche, Cateland, Gilbert, Peitgen, Shallit, and Skordev.

cs.FL↗