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Giedrius Alkauskas

Publications and source records attributed to Giedrius Alkauskas.

At least 19 recordsLinked to original sources
arXiv2026

Squares, three fleas, sporadic sets, and squares

Three fleas are on the plane. At each move, two fleas jump to the other two vertices of a square erected on the segment joining them. Starting from $\{(0,0),(2,1),(3,2)\}$, we ask which areas can occur for the triangle they span. Every positive half-integer occurs, and the attainable integer areas meet every residue class modulo every modulus. Yet no perfect square ever occurs. We explain this unexpected obstruction using quadratic reciprocity, and then uncover its geometric source: the flea dynamics are closely related to integral Apollonian circle packings. An exact computation up to $10^{12}$ finds only five additional missing nonsquares: $5,29,80,99,179$.

math.NT↗
arXiv2026

Regular triangle unions with maximal number of sides

Fix an integer n>=1. Suppose that a simple polygon is the union of n triangles whose vertices along the common boundary are arranged cyclically. How many sides can such a union -- to be called regular -- have at most? This gives OEIS sequence A375986, a recent entry. It will be shown here that the sequence begins 3, 12, 22, 33, 45, 56, 67, 80, 91, and satisfies linear lower and upper bounds. The latter is not merely an estimate: it is realizable combinatorially. This leads to two further questions: can the same combinatorics be realized in pseudoline geometry, and if so, can such a realization be stretched? The paper is largely expository, with excursions into neighboring topics (union complexity, the Zone Theorem, stretchability, the Kobon triangle problem, Davenport-Schinzel sequences, lower envelopes of line segments). However, it adds a new tool tailored for studying regular unions; namely, triangulation shifts. In essence, this is a method to represent any such n-union by a triangulation of a regular (n+1)-gon and its dynamical mutation.

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arXiv2024

Full Grid Lattice Polygons with Maximal Sum of Squares of Edge-Lengths

Consider a subset [1,2,...,n]x[1,2,...,n] of the plane integer lattice. Take any non self-intersecting n^2-gon built on it (straight angles are allowed). The square of a side length is a positive integer. It is thus natural to ask how large the sum of square lengths of such an n^2-gon can be. This maximal value is a new integer sequence, labeled by A358212 in OEIS. In this note we give the lower bound and conjecture that this in fact is the correct answer. We further investigate proper n^2-gons (straight angles are not allowed) and present analogous results. Both sequences (conjecturally) have a different growth size.

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arXiv2024

Friendly paths for finite subsets of plane integer lattice. I

For a given finite subset P of points of the lattice Z^2, a friendly path is a monotone (uphill or downhill) lattice path which splits points in half; points lying on the path itself are discarded. The purpose of this paper (and its sequel) is to fully describe all configurations of n points in Z^2 which do not admit a friendly path. We say that such an n-set is inseparable. There are, up to the lattice symmetry, exactly c(n) such sets. If only lattice shifts are counted, there are ĉ(n) of them. Both sequences are new entries into OEIS (A369382 and, respectively, A367783). In particular, n=27 is the first odd numbers with c(n)=1. No example was known so far. This solves problem 11484(b)* posed in American Mathematical Monthly (February 2010). In this paper we also show that inseparable n-set exist for all even numbers n>=12 and almost all odd numbers.

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arXiv2018

Planar 2-homogeneous commutative rational vector fields

In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In the latter case, orbits of each flow are given in terms of $1$-homogeneous rational functions $W$ as curves $W(x,y)=\textrm{const}$. An exhaustive method to construct such commuting algebraic flows is presented. The degree of the so-obtained algebraic functions in two variables can be arbitrarily high.

math.AG↗
arXiv2018

Projective and polynomial superflows. I

Let $x\in\mathbb{R}^{n}$. For $ϕ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $ϕ^{t}=t^{-1}ϕ(xt)$. A projective flow is a solution to the projective translation equation $ϕ^{t+s}=ϕ^{t}\circϕ^{s}$, $t,s\in\mathbb{R}$. Previously we have developed an arithmetic, topologic and analytic theory of $2$-dimensional projective flows: rational, algebraic, unramified, abelian flows, commuting flows. The current paper is devoted to highly symmetric flows - superflows. Within flows with a given symmetry, superflows are unique and optimal. Our first result classifies all $2$-dimensional superflows. For any positive integer $d$, there exists the superflow $ϕ_{\mathbb{D}_{2d+1}}$ whose group of symmetries is the dihedral group $\mathbb{D}_{2d+1}$. In the current paper we explore the superflow $ϕ_{\mathbb{D}_{5}}$, which leads to investigation of abelian functions over curve of genus $6$. The $3$-dimensional theory of projective flows is more involved. We investigate two different $3$-dimensional superflows, whose group of symmetries are, respectively, the full tetrahedral group $\widehat{\mathbb{T}}$ (all symmetries of a tetrahedron), and the octahedral group $\mathbb{O}$ (orientation preserving symmetries of an octahedron), both isomorphic, though non-equivalent as representations. The generic orbits of the first flow are space curves of genus $1$, and the flow itself can be analytically described in terms of Jacobi elliptic functions. The generic orbits of the second flow are curves of genus $9$, and the flow itself can be described in terms of Weierstrass elliptic functions (via reduction of hyper-elliptic functions to elliptic). In the second part of this work we will classify all $3$-dimensional superflows (including the icosahedral superflow), and in the third we investigate superflows over $\mathbb{C}$.

math.AG↗
arXiv2018

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove an asymptotic formula for the eigenvalues of L. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.

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arXiv2017

Beltrami vector fields with polyhedral symmetries

A 3-dimensional vector field $B$ is said to be Beltrami vector field (force free-magnetic vector field in physics), if $B\times(\nabla\times B)=0$. Motivated by our investigations on projective an polynomial superflows, and as an important side result, in the first paper on this topic we constructed two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and that both have orientation-preserving icosahedral symmetry (group of order $60$). In the current paper we extend these results to the tetrahedral and octahedral cases, and (together with an icosahedral case) we calculate all simplest Beltrami fields with polyhedral symmetries arising from solutions to the Helmholtz equation of any order (the first aforementioned paper being an order 1 approach). The notion of Beltrami vector field, slightly relaxed, generalizes to any dimension. In this paper we also present 2-dimensional vector fields which have a dihedral symmetry $\mathbb{D}_{2d+1}$ of order $4d+2$. A much more detailed analysis is carried out in case $d=1$. One of these fields is particularly exceptional since it is the only case in our investigations which arises from the order $0$ approach to the Helmholtz equation, thus relating this flow to the $ABC$ flow.

math.CA↗
arXiv2017

Beltrami vector fields with an icosahedral symmetry

A vector field is called a Beltrami vector field, if $B\times(\nabla\times B)=0$. In this paper we construct two unique Beltrami vector fields $\mathfrak{I}$ and $\mathfrak{Y}$, such that $\nabla\times\mathfrak{I}=\mathfrak{I}$, $\nabla\times\mathfrak{Y}=\mathfrak{Y}$, and such that both have an orientation-preserving icosahedral symmetry. Both of them have an additional symmetry with respect to a non-trivial automorphism of the number field $\mathbb{Q}(\,\sqrt{5}\,)$.

math.DG↗
arXiv2017

The modular group and words in its two generators

Consider the full modular group $\sf{PSL}_{2}(\mathbb{Z})$ with presentation $\langle U,S|U^3,S^2\rangle$. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper might be considered as a necessary appendix), we are lead to the following natural question. Some words in the alphabet $\{U,S\}$ are equal to the unity; for example, $USU^3SU^2$ is such a word of length $8$, and $USU^3SUSU^3S^3U$ is such a word of length $15$. Given $n\in\mathbb{N}_{0}$. Find the number of words of length $n$ which are equal to the unity. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over $\mathbb{Q}(x)$ of degree $3$. As an aside, we formulate the problem of describing all algebraic functions with a Fermat property.

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arXiv2016

The projective translation equation and rational plane flows. II. Corrections and additions

In this second part of the work, we correct the flaw which was left in the proof of the main Theorem in the first part. This affects only a small part of the text in this first part and two consecutive papers. Yet, some additional arguments and additions are needed to claim the validity of the classification results. With these new results in a disposition, algebraic and rational flows can be much more easily and transparently classified. It also turns out that the notion of an algebraic projective flow is a very natural one. For example, we give an inductive (on dimension) method to build algebraic projective flows with rational vector fields, and ask whether these account for all such flows. Further, we expand on results concerning rational flows in dimension $2$. Previously we found such flows symmetric with respect to a linear involution $i(x,y)=(y,x)$. Here we find all rational flows symmetric with respect to a non-linear $1$-homogeneous involution $i(x,y)=(\frac{y^2}{x},y)$. We also find all solenoidal rational flows. Up to linear conjugation, there appears to be exactly two non-trivial examples.

math.CA↗
arXiv2016

Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems

In this note we list a number of open problems in the fields of number theory, combinatorics, and representation theory: algebraic functions with Fermat property; power product expansion of the generating function for the partition function; relation between the non-trivial Riemann zeros and eigenvalues of the transfer operator; functional equation related to norm forms; two problems from geometric combinatorics; a problem on the moments of the Minkowski question mark function; a question in representation theory; a problem on interpolating the moments of the Stern's diatomic sequence; an arithmetic properties of the binary composition function.

math.NT↗
arXiv2016

Projective superflows. II. $O(3)$ and the icosahedral group

Let $X\in\mathbb{R}^{n}$. For $ϕ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $ϕ^{t}=t^{-1}ϕ(Xt)$. A projective flow is a solution to the projective translation equation $ϕ^{t+s}=ϕ^{t}\circϕ^{s}$, $t,s\in\mathbb{R}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is in a sense unique and optimal. In this second part we classify $3$-dimensional real superflows. Apart from the superflow $ϕ_{\hat{\mathbb{T}}}$ (with a group of symmetries being all symmetries of a tetrahedron) and the superflow $ϕ_{\mathbb{O}}$ (with a group of symmetries being orientation preserving symmetries of an octahedron), both described in the first part of this study, here we investigate in detail the superflow $ϕ_{\mathbb{I}}$ whose group of symmetries is the icosahedral group $\mathbb{I}$ of order $60$. This superflow is a flow on co-centric spheres, and is also solenoidal. These three superflows is the full (up to linear conjugation) list of $3$-dimensional irreducible real projective superflows. We also find all reducible $3$-dimensional real superflows. There are two of them: one with group of symmetries being all symmetries of a $3$-prism (group of order $12$), and the second with a group of symmetries being all symmetries of a $4$-antiprism (group of order $16$).

math.AG↗
arXiv2016

Projective superflows. III. Finite subgroups of $U(2)$

Let $X\in\mathbb{R}^{n}$ or $\mathbb{C}^{n}$. For $ϕ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ (respectively, $ϕ:\mathbb{C}^{n}\mapsto\mathbb{C}^{n}$) and $t\in\mathbb{R}$ (respectively, $\mathbb{C}$), we put $ϕ^{t}=t^{-1}ϕ(Xt)$. A projective flow is a solution to the projective translation equation $ϕ^{t+s}=ϕ^{t}\circϕ^{s}$, $t,s\in\mathbb{R}$ or $\mathbb{C}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is, up to a homothety, unique and optimal. In the first and the second part of this work we classified real $2$ and $3-$dimensional supeflows over $\mathbb{R}$. In this third part we classify all $2-$dimensional complex superflows; that is, whose group of symmetries are finite subgroups of $U(2)$. This includes both irreducible and reducible superflows.

math.AG↗
arXiv2016

Algebraic and abelian solutions to the projective translation equation

Let $\mathbf{x}=(x,y)$. A projective 2-dimensional flow is a solution to a 2-dimensional projective translation equation (PrTE) $(1-z)ϕ(\mathbf{x})=ϕ(ϕ(\mathbf{x}z)(1-z)/z)$, $ϕ:\mathbb{C}^{2}\mapsto\mathbb{C}^{2}$. Previously we have found all solutions of the PrTE which are rational functions. The rational flow gives rise to a vector field $\varpi(x,y)\bullet\varrho(x,y)$ which is a pair of 2-homogenic rational functions. On the other hand, only very special pairs of 2-homogenic rational functions, as vector fields, give rise to rational flows. The main ingredient in the proof of the classifying theorem is a reduction algorithm for a pair of 2-homogenic rational functions. This reduction method in fact allows to derive more results. Namely, in this work we find all projective flows with rational vector fields whose orbits are algebraic curves. We call these flows abelian projective flows, since either these flows are parametrized by abelian functions and with the help of 1-homogenic birational plane transformations (1-BIR) the orbits of these flows can be transformed into algebraic curves $x^{A}(x-y)^{B}y^{C}\equiv\mathrm{const.}$ (abelian flows of type I), or there exists a 1-BIR which transforms the orbits into the lines $y\equiv\mathrm{const.}$ (abelian flows of type II), and generally the latter flows are described in terms of non-arithmetic functions. Our second result classifies all abelian flows which are given by two variable algebraic functions. We call these flows algebraic projective flows, and these are abelian flows of type I. We also provide many examples of algebraic, abelian and non-abelian flows.

math.AG↗
arXiv2015

The Minkowski $?(x)$ function, a class of singular measures, quasi-modular and mean-modular forms. I

The Minkowski question mark function is a rich object which can be explored from the perspective of dynamical systems, complex dynamics, metric number theory, multifractal analysis, transfer operators, integral transforms, and as a function itself via analysis of continued fractions and convergents. Our permanent target, however, was to get arithmetic interpretation of the moments of ?(x) (which are relatives of periods of Maass wave forms) and to relate the function ?(x) to certain modular objects. In this paper we establish this link, embedding ?(x) not into the modular-world itself, but into a space of functions which are generalizations and which we call mean-modular forms. For this purpose we construct a wide class of measures, and also investigate modular forms for congruence subgroups which additionally satisfy the three term functional equation. From this perspective, the modular forms for the whole modular group as well as the Stieltjes transform of ?(x) (the dyadic period function) minus the Eisenstein series of weight 2 fall under the same uniform definition. The main result is the construction of the canonical isomorphism between the spaces of quasi-modular forms and mean-modular forms. This gives unexpected Minkowski question mark function-related interpretation of quasi-modular forms.

math.NT↗
arXiv2013

The projective translation equation and rational plane flows. I

Let X=(x,y). A plane flow is a function F(X,t): R^2*R->R^2 such that F(F(X,s),t)=F(X,s+t) for (almost) all real numbers x,y,s,t (the function F might not be well-defined for certain x,y,t). In this paper we investigate rational plane flows which are of the form F(X,t)=f(Xt)/t; here f is a pair of rational functions in 2 real variables. These may be called projective flows, and for a description of such flows only the knowledge of Cremona group in dimension 1 is needed. Thus, the aim of this work is to completely describe over R all rational solutions of the two dimensional translation equation (1-z)f(X)=f(f(Xz)(1-z)/z). We show that, up to conjugation with a 1-homogenic birational plane transformation (1-BIR), all solutions are as follows: a zero flow, two singular flows, an identity flow, and one non-singular flow for each non-negative integer N, called the level of the flow. The case N=0 stands apart, while the case N=1 has special features as well. Conjugation of these canonical solutions with 1-BIR produce a variety of flows with different properties and invariants, depending on the level and on the conjugation itself. We explore many more features of these flows; for example, there are 1, 4, and 2 essentially different symmetric flows in cases N=0, N=1, and N>=2, respectively. Many more questions will be treated in the second part of this work.

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