Additive Combinatorics

We investigate a question posed by Diaconis & Fulman (2023) regarding modular in-shuffling and the distribution of card positions in binary form. Specifically, we consider the sequence 𝑋 𝑘 , which records whether the position of a tracked card lies in the top half of a deck after 𝑘 iterations of the modular in-shuffle map 𝐼(𝑗) = 2𝑗 mod 𝑝, where 𝑝 is a is a prime for which 2 is a primitive root. We prove that the random sequence {𝑋 𝑘 } is equidistributed and asymptotically behaves like a sequence of independent and identically distributed Bernoulli(1 2) random variables. We provide both theoretical justification and empirical evidence with computed values of 𝑋 𝑘 for large 𝑝 supporting this behavior.

This work introduces and studies a special class of subsets of infinite increasing sequences, called oscillating subsets, characterized by bounded spacing between their positions in the original set. The central result demonstrates that these structurally regular subsets preserve the convergence or divergence behavior of the harmonic series formed from the original sequence. This establishes a robust link between the structural arrangement of elements and the analytic behavior of their reciprocals, offering a new perspective on how local regularity governs global summation properties.

This study presents a one-step multi-derivative hybrid block method (OSMDHBM) of order ten, which incorporates third derivatives for the solution of linear and nonlinear second-order initial value problems (IVPs). The derivation incorporates a multi-step collocation and interpolation method, using an approximated power series as the basis function. The intra-step or off-step points are obtained from the derivative of a shifted Legendre polynomial (SLP) of degree four. The accuracy, consistency, and stability properties of the method are analyzed. The nonlinear IVPs are linearized using the modified Picard iteration method (MPIM). In order to demonstrate the superiority of the method, numerical experiments are presented. Comparisons are made between the numerical results obtained and results from other methods and similar schemes in the literature.

A finite set of integers $A$ is a sum-dominant (also called an More Sums Than Differences or MSTD) set if $|A+A| > |A-A|$. While almost all subsets of $\{0, \dots, n\}$ are not sum-dominant, interestingly a small positive percentage are. We explore sufficient conditions on infinite sets of positive integers such that there are either no sum-dominant subsets, at most finitely many sum-dominant subsets, or infinitely many sum-dominant subsets. In particular, we prove no subset of the Fibonacci numbers is a sum-dominant set, establish conditions such that solutions to a recurrence relation have only finitely many sum-dominant subsets, and show there are infinitely many sum-dominant subsets of the primes.

Let G be a prolific graph, by which we mean a finite connected simple graph which is not isomorphic to a cycle nor a path nor the star graph K 1,3. The line-graph of G, denoted by L(G), is defined by having its vertex-set equal to the edge-set of G and two vertices of L(G) are adjacent if the corresponding edges are adjacent in G. For a positive integer k, the iterated line-graph L k (G) is defined recursively by L k (G) = L(L k−1 (G)). In this paper we shall consider fifteen well-known graph parameters and study their behaviour when the operation of taking the line-graph is iterated. We shall first show that all of these parameters are unbounded, that is, if P = P (G) is such a parameter defined on any prolific graph G, then P (L k (G)) → ∞ when k → ∞. This idea of unboundedness is motivated by a well-known old result of van Rooij and Wilf that says that the number of vertices is unbounded if and only if the graph is prolific. Following this preliminary result, the main thrust of the paper will be the study of the value of k(P, F), which is the index of a family of prolific graphs with regards to a given graph parameter P (G). For a given parameter P (G), the index of G is denoted by ind(P, G) = min{r : P (G) < P (L r (G)}. Now for a family F of prolific graphs, the index of the family is k(P, F) = max{ind(P, G) : G ∈ F}, that is k(P, F) is the smallest integer k such that for every prolific graphs G ∈ F, ind(P, G) ≤ k(P, F). The problem of determining the index of a parameter over the family of prolific graphs is motivated by a classical result of Chartrand who showed that it could require k = |V (G)| − 3 iterations to guarantee that L k (G) has a hamiltonian cycle. For twelve of the fifteen parameters considered, we exactly determine k(P, F) where F is the family of all prolific graphs, and for some parameters we also characterize the class of prolific graphs realizing the extremal value k(P, F). For example, for the matching number µ, we show that the index of every prolific graph is at most 4 which is sharp, namely k(µ, F) = 4 and we further characterize those graphs for which ind(µ, G) = 4. Interesting open problems remain, in particular completing the determination of k(P, F) for the three parameters: the independence number, independent domination number and domination number, where we obtain partial results.

Let G be a prolific graph, by which we mean a finite connected simple graph which is not isomorphic to a cycle nor a path nor the star graph K 1,3. The line-graph of G, denoted by L(G), is defined by having its vertex-set equal to the edge-set of G and two vertices of L(G) are adjacent if the corresponding edges are adjacent in G. For a positive integer k, the iterated line-graph L k (G) is defined recursively by L k (G) = L(L k−1 (G)). In this paper we shall consider fifteen well-known graph parameters and study their behaviour when the operation of taking the line-graph is iterated. We shall first show that all of these parameters are unbounded, that is, if P = P (G) is such a parameter defined on any prolific graph G, then P (L k (G)) → ∞ when k → ∞. This idea of unboundedness is motivated by a well-known old result of van Rooij and Wilf that says that the number of vertices is unbounded if and only if the graph is prolific. Following this preliminary result, the main thrust of the paper will be the study of the value of k(P, F), which is the index of a family of prolific graphs with regards to a given graph parameter P (G). For a given parameter P (G), the index of G is denoted by ind(P, G) = min{r : P (G) < P (L r (G)}. Now for a family F of prolific graphs, the index of the family is k(P, F) = max{ind(P, G) : G ∈ F}, that is k(P, F) is the smallest integer k such that for every prolific graphs G ∈ F, ind(P, G) ≤ k(P, F). The problem of determining the index of a parameter over the family of prolific graphs is motivated by a classical result of Chartrand who showed that it could require k = |V (G)| − 3 iterations to guarantee that L k (G) has a hamiltonian cycle. For twelve of the fifteen parameters considered, we exactly determine k(P, F) where F is the family of all prolific graphs, and for some parameters we also characterize the class of prolific graphs realizing the extremal value k(P, F). For example, for the matching number µ, we show that the index of every prolific graph is at most 4 which is sharp, namely k(µ, F) = 4 and we further characterize those graphs for which ind(µ, G) = 4. Interesting open problems remain, in particular completing the determination of k(P, F) for the three parameters: the independence number, independent domination number and domination number, where we obtain partial results.

A \(t\)-AP is a sequence of the form \(a,a+d,\ldots, a+(t-1)d\), where \(a,d\in \mathbb{Z}\). Given a finite set \(X\) and positive integers \(d\), \(t\), \(a_1,a_2,\ldots,a_{t-1}\), define \(\nu(X,d) = |\{(x,y):{x,y\in{X}},{y&gt;x},{y-x=d}\}|\), \((a_1,a_2,\ldots,a_{t-1};d) =\) a collection \(X\) s.t. \(\nu(X,d\cdot{i})\geq a_i\) for \(1\leq i\leq t-1\). In this paper, we investigate the structure of sets with bounded number of pairs with certain gaps. Let \((t-1,t-2,\ldots,1; d)\) be called a \emph{\(t\)-AP distance-set} of size at least \(t\). A \(k\)-colouring of integers \(1,2,\ldots, n\) is a mapping \(\{1,2,\ldots,n\}\rightarrow \{0,1,\ldots,k-1\}\) where \(0,1,\ldots,k-1\) are colours. Let \(ww(k,t)\) denote the smallest positive integer \(n\) such that every \(k\)-colouring of \(1,2,\ldots,n\) contains a monochromatic \(t\)-AP distance-set for some \(d&gt;0\). We conjecture that \(ww(2,t)\geq t^2\) and prove the lower bound for most cases. We also generalize the notion of \...

We prove a generalization of the Droz-Farny line theorem with or-thologic triangles. In 1899, Arnold Droz-Farny [3] discovered and published without proof the following beautiful result. Theorem 1 (Droz-Farny line theorem). Two perpendicular lines 1 and 2 pass through the orthocenter of triangle ABC. If 1 intersects BC, CA, AB at D 1 , E 1 , F 1 , and 2 intersects BC, CA, AB at D 2 , E 2 , F 2 , then the midpoints of D 1 D 2 , E 1 E 2 , F 1 F 2 are collinear. The line containing these midpoints is called the Droz-Farny line associated with 1 and 2. There are a number of generalizations of the Droz-Farny line theorem, notably by van Lamoen [5] (see also [4, 8]), Thas, [10], and Bradley [2]. The most recent one is by Letrouit [6]. If directly similar triangles DD 1 D 2 , EE 1 E 2 , F F 1 F 2 are constructed on D 1 D 2 , E 1 E 2 , F 1 F 2 respectively, then D, E, F are collinear (see also Nicollier [7]). In this note we prove a generalization associated with a pair of orthologic triang...

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science.

We colour every point x of a probability space X according to the colours of a finite list x 1 , x 2 ,. .. , x k of points such that each of the x i , as a function of x, is a measure preserving transformation. We ask two questions about a colouring rule: (1) does there exist a finitely additive extension of the probability measure for which the x i remain measure preserving and also a colouring obeying the rule almost everywhere that is measurable with respect to this extension?, and (2) does there exist any colouring obeying the rule almost everywhere? If the answer to the first question is no and to the second question yes, we say that the colouring rule is paradoxical. A paradoxical colouring rule not only allows for a paradoxical partition of the space, it requires one. We pay special attention to generalizations of the Hausdorff paradox.

We show how the minimal free resolution of a set of n points in general position in projective space of dimension n − 2 explicitly determines structure constants for a ring of rank n. This generalises previously known constructions of Levi-Delone-Faddeev and Bhargava in the cases n = 3, 4, 5. 2. Statement of the main theorem We recall a few basic notions from commutative algebra that will be needed to state our main theorem. Throughout, we work over a field K with algebraic closureK. Let R = K[x 1 ,. .. , x m ] be the polynomial ring with its usual grading. For M = ⊕ d M d a graded R-module, we write M(c) = ⊕ d M c+d for the graded R-module with grading shifted by c. A direct sum of modules of the form R(c) is a called a graded free R-module. Definition 2.1. A graded free resolution of a graded R-module M is a chain complex F • of

We show how the minimal free resolution of a set of n points in general position in projective space of dimension n − 2 explicitly determines structure constants for a ring of rank n. This generalises previously known constructions of Levi-Delone-Faddeev and Bhargava in the cases n = 3, 4, 5.

Numbers similar to those of van der Waerden are examined. We consider increasing sequences of positive integers {x,.x2,. .. . x,} either that form an arithmetic sequence or for which there exists a polynomial f(x) = x;;$ a,2 with a,EZ, a,-2>0, and x,+,= J(.x,). We denote by q(n) the least positive integer such that if { 1, 2,. .. . q(n)} is 2-colored, then there exists a monochromatic sequence of the type just described. We give an upper bound for q(n), as well as values of q(n) for n d 5. A stronger upper bound for q(n) is conjectured and is shown to imply the existence of a similar bound on the nth van der Waerden number.

In 1983, Bouchet proposed a conjecture that every flow-admissible signed graph admits a nowhere-zero 6-flow. Bouchet himself proved that such signed graphs admit nowhere-zero 216-flows and Zýka further proved that such signed graphs admit nowherezero 30-flows. In this paper we show that every flow-admissible signed graph admits a nowhere-zero 11-flow.

Let K be a field and R = K[x 1 ,. .. , x n ]. We obtain an improved upper bound for asymptotic resurgence of squarefree monomial ideals in R. We study the effect on the resurgence when sum, product and intersection of ideals are taken. We obtain sharp upper and lower bounds for the resurgence and asymptotic resurgence of cover ideals of finite simple graphs in terms of associated combinatorial invariants. We also explicitly compute the resurgence and asymptotic resurgence of cover ideals of several classes of graphs. We characterize a graph being bipartite in terms of the resurgence and asymptotic resurgence of edge and cover ideals. We also compute explicitly the resurgence and asymptotic resurgence of edge ideals of some classes of graphs.

Let D = (G, O, w) be a weighted oriented graph whose edge ideal is I(D). In this paper, we characterize the unmixed property of I(D) for each one of the following cases: G is an SCQ graph; G is a chordal graph; G is a simplicial graph; G is a perfect graph; G has no 4-or 5-cycles; G is a graph without 3-and 5-cycles; and girth(G) 5.

A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.

A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.

A famous theorem of Szemerédi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial (and graph-theoretical) approach of Szemerédi, the ergodic theory approach of Furstenberg, the Fourier-analytic approach of Gowers, and the hypergraph approach of Nagle-Rödl-Schacht-Skokan and Gowers. In this lecture series we introduce the first, second and fourth approaches, though we will not delve into the full details of any of them. One of the themes of these lectures is the strong similarity of ideas between these approaches, despite the fact that they initially seem rather different.

Let A be an infinite sequence of positive integers a, < a, < .I. and put fAxI = CnsR,oCx (l/a), D,(x) = maxtcnGx CoeA,o,n 1. In Part I, it was proved that limx++m supD,(x)/f,(x) = +co. In this paper, this theorem is sharpened by estimating DA(x) in terms of f,(x). It is shown that lim X+ +a3 sup DA(x) exp(-c,(logf,(x))') = +a~ and that this assertion is not true if c, is replaced by a large constant c2.

Let p be the characteristic of F q and let q be a primitive root modulo a prime r = 2n + 1. Let β ∈ F q 2n be a primitive rth root of unity. We prove that the multiplicative order of the Gauss period β + β −1 is at least (log p) c log n for some c > 0. This improves the bound obtained by Ahmadi, Shparlinski and Voloch when p is very large compared with n. We also obtain bounds for "most" p.

Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, with $C(f)$ exceptions, the solutions $(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where $c$ is a constant and ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline {\mathbb F}_p^*$. Moreover, for most $p\lt N$, $N$ being a large number, we prove that, with $C(f)$ exceptions, ${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where $\epsilon (p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty $.