Papers

We revisit Zeilberger's computational framework for enumerating polyomino tilings and demonstrate its use in counting tilings of $k\times n$ boards by $L$-tetrominoes, allowing rotations. Recently, Bělohoubek and Slavík gave a generating function for tilings of $2n\times 4$ boards by $L$-tetrominoes. Motivated by their work, we first verify the method by automatically recovering their generating function. We then apply the method to widths $5$ through $9$ and obtain explicit rational generating functions and new integer sequences that are not currently listed in the OEIS.

We investigate manifestations of the Nilpotent Heuristic, which posits that recurrence and convergence phenomena known for measure-preserving $\mathbb{Z}^D$-systems extend to nilpotent group actions. Our main results establish seminorm estimates and limiting formulas for multiple ergodic averages arising from actions of 2-step nilpotent groups. In particular, if $T_1,\ldots,T_\ell$ are totally ergodic and generate a 2-step nilpotent group, then \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T_1^n f_1 \cdots T_\ell^{n^\ell}f_\ell = \prod_{j=1}^{\ell}\int f_j\,d\mu \] in the $L^{2}$ norm for all bounded functions $f_{1},\dots,f_{\ell}$; the same holds for any distinct-degree polynomial iterates. We also obtain popular-common-difference versions of the polynomial Szemeédi theorem in the same setting. In a different direction, our approach allows us to completely resolve the joint ergodicity conjecture for multidimensional polynomials and $\mathbb Z^D$-systems; we also present an example showing that, surprisingly enough, the 2-step nilpotent analog fails. We conclude with many open problems concerning joint ergodicity, seminorm estimates, and the structure theory of nilpotent systems.

Given $r \in \mathbb{N}$ and an edge-coloured complete graph such that the average number of colours incident with a vertex is at most $r$, Conlon and Stein asked whether there is a vertex partition into a bounded number of monochromatic cycles, and proved this for $r=2$. We settle the first open case by proving the corresponding statement for $r=3$.

The independence polynomial of a simple graph $G$ is given by \( I_G(z) = i_0 + i_1 z + i_2 z^2 + \cdots + i_\alpha z^\alpha \), where \( i_\alpha \) denotes the size of a maximum independent set, also called the independence number of the graph. The independence polynomial has the notable feature of being essentially closed under graph composition (lexicographic product). In this paper, we determine the independence polynomials of size five. For a disconnected graph $G$, we exploit the fact that $I_G(z)$ factors as the product of the independence polynomials of the connected components of $G$. Furthermore, we classify all independence polynomials that can occur for such a disconnected graph $G$ and, by examining their component structures, we characterize the disconnected configurations that may arise.

Let $S$ be a nonempty finite set of positive integers, let $q=\max S$, and let $B_n(S)$, $n>2q$, be the normalized product sequence associated with the Chebyshev-type polynomial of the cyclic circulant graph $G_n(S)$. When $G_n(S)$ is connected, $B_n(S)$ is both the normalized spanning-tree number and a normalized special value of the reduced spectral function, a determinant-type function constructed from the non-trivial adjacency spectrum. Chebyshev root representations and the existence of linear recurrences for fixed-step circulant spanning-tree sequences are known. Starting from these representations, we explicitly construct a monic annihilating polynomial $\mathcal H_S(X)\in\mathbb Z[X]$ of degree $3^{q-1}$, which yields a general upper bound for the recurrence order of $B_n(S)$. By collecting coincident exponential bases and accounting for possible cancellations, we determine the minimal annihilating polynomial and give a sufficient condition under which the minimal recurrence order is exactly $3^{q-1}$. For $S=\{1,2,3\}$ and $S=\{1,3\}$, we explicitly derive the corresponding ninth-degree annihilating polynomials and prove their minimality.

A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. This model is used to capture the idea of vertices having roles in a contact network, consistent with the set of roles of their contacts. We define an extension of the role colouring problem to temporal graphs. Temporal roles are defined via an automaton with states and transitions capturing both the current colour of a vertex and information about its current and past adjacencies. We show, by a reduction from the static problem, that the temporal role colouring problem is NP-complete. To contend with this intractability, we explore several parameterisations. We give fixed-parameter tractability results with respect to the number of states of the automaton combined with either the vertex-interval-membership width or the tree-interval-membership width of the temporal graph. We further show the problem is in FPT parameterised by the treewidth of the underlying graph, the lifetime of the temporal graph and the number of colours combined.

In this note, we study two simple operations for extending juggling (siteswap) sequences, called first throws and last catches. Both constructions come from a natural question for a juggler: how can one add throws at the beginning or catches at the end of a pattern without creating a collision? Using landing times and the permutation test, we give necessary and sufficient conditions under which these constructions, when applied to an arbitrary valid siteswap, produce another valid siteswap. We complement our analysis with several examples. An interactive visualization of these extensions is available in [Para].

We describe algorithms for the exhaustive enumeration and classification of simple arrangements of $n$ pseudolines ($n$ odd) maximizing the number of triangular faces. The depth-first search enumerates reduced words for the longest permutation $w_0$ by branching only on the even-indexed generators, using pruning constraints imposed by the geometry of optimal arrangements. The approach handles both perfect arrangements with a regular triangular pattern and unavoidable deviations from it for $n \equiv 1 \pmod 6$. The output is classified into a hierarchy of equivalence classes: by commutation, by Euclidean transformations, and by projective transformations. For each projective class we recover its full symmetry group $G \subseteq S_{n+1}$ together with the orbit-stabilizer profile of its Euclidean subclasses. Completeness of the search and classification is proved: every wiring diagram is reached. We report full enumerations; e.g. for $n=27$, 85,562,064 wiring diagrams partitioned into 56,646 projective classes. For larger $n$ (up to $n=93$), where exhaustive enumeration is out of reach, we report partial (first-hit) results.

Textile systems are best known to model two-dimensional shifts of finite type. In this article, we associate a discrete algebra with a textile system and provide a groupoid model for it. When the textile system is left-resolving, this algebra coincides with the Kumjian–Pask algebra of the associated $2$-graph. The main objective of this paper is to extend textile systems to dimension $3$ so that the resulting structures can, on the one hand, capture all three-dimensional shifts of finite type and, on the other hand, provide a textile-like framework for $3$-graphs extending the well-known connection between $2$-graphs and left-resolving textile systems. We introduce a model of a three-dimensional textile system and investigate the interplay between such textile systems and $3$-graphs. In particular, our investigation shows that the conditions required to form a $3$-graph from a $3$-colored graph (including the delicate associativity condition on tricolored paths), can be encoded in terms of simple pullback diagrams arising from the textile data. We also define homology groups for three-dimensional textiles and prove that these groups coincide with the homology groups of the associated $3$-graphs, thus establishing that our construction is homologically consistent with $3$-graphs.

We present a unified method for evaluating finite sums of powers of cotangent, tangent, cosecant and secant weighted by primitive $k$th roots of unity. The approach relies on constant term extraction for iterated Laurent series, combined with generating functions and partial fraction decomposition. We obtain explicit closed-form expressions for all even-power sums of the four functions, as well as for odd-power cotangent and tangent sums. The formulas are given in terms of Bernoulli polynomials, Euler polynomials, and universal coefficients $r_{n,t}$. As applications, we recover numerous classical identities—including the ordinary and alternating cotangent power sums and Acton's alternating tangent sum—in a systematic and elementary way.

Let $r\ge 2$ be a real number, $d$ a positive integer. A $d$-dimensional nowhere-zero $r$-flow, or $(r,d)$-NZF, on a graph $G$ is an orientation of $G$ together with a function $f\colon E(G)\to \mathbb{R}^d$, such that for all $e\in E(G)$, the Euclidean norm of $f(e)$ lies in the interval $[1,r-1]$, and for every $v\in V(G)$ the sum of all incoming flow values at $v$ equals the sum of all outgoing ones. The $d$-dimensional flow number of $G$ is the parameter $\phi_d(G)=\inf \{r\colon G$ has an $(r,d)$-NZF$\}$. In this paper we provide a lower bound for the $2$-dimensional flow number of the the Flower snark. In particular, together with a previous numerical result by the authors, we prove that $\phi_2(J_n) \in [1 + 2 \sin\frac{5}{22}\pi, 2.387893647]$, where $J_n$ denotes the Flower snark on $4n$ vertices.

Let $X,X'$ be independent and identically distributed discrete real-valued random variables of finite Shannon entropy, and write $H(X)$ for the Shannon entropy of $X$. We prove that \[ \max\{H(X+X'),\,H(XX')\} \ge \frac87 H(X)-O(\log H(X)). \] This is the entropic analog of the celebrated sum-product phenomenon, and answers a question of Goh, which simply asked for a coefficient strictly larger than 1. An example by the author, Gavalakis, and Kontoyiannis showed the coefficient cannot exceed $\frac43$. Previous work by Gavalakis, Goh, and Kontoyiannis was able to prove a result of a weaker form, which could not translate to a coefficient strictly larger than 1 because of examples where the min-entropy is significantly smaller than the Shannon entropy. By splitting the distribution of $X$ into uniform pieces, which costs $O(\log H(X))$ entropy, we obviate this issue, establishing a coefficient of $\frac{10}{9}$. We augment this to $\frac87$ by adapting the work of Solymosi, which established the combinatorial sum-product phenomenon with coefficient $\frac43$ by bounding the multiplicative energy, to the entropy setting, again via a uniformization technique.

Aldous' spectral gap conjecture states that the random walk and the interchange process on any connected graph have the same spectral gap, or, equivalently, the second largest eigenvalue of any connected Cayley graph on the symmetric group $S_n$ with respect to a set of transpositions is achieved by the standard representation of $S_n$. This celebrated conjecture, proved in its general form in 2010, has inspired much interest in searching for other Cayley graphs on $S_n$ possessing this property, now known as the Aldous property. In this paper, we first prove that for $n \ge 5$ at most one of a normal Cayley graph on $S_n$ and its complement can possess the Aldous property except when these two graphs are $2K_{n!/2}$ and $K_{n!/2,n!/2}$ respectively. We then determine, for sufficiently large $n$, all normal Cayley graphs $\mathrm{Cay}(S_n, S)$ that have the Aldous property, except for the case when $S$ contains a permutation with support size in $\{2, 3, \dots, n-2\}$ and a permutation with support size in $\{n-1, n\}$, but not all permutations with support size $n$ are contained in $S$. In particular, we show that a non-complete normal Cayley graph $\mathrm{Cay}(S_n, S)$ does not have the Aldous property if all permutations in $S$ have support size $n-1$ or $n$, or all permutations with support size $n$ are contained in $S$, thereby solving an open problem posed by Li, Xia and Zhou in 2023. Along the way we determine all normal Cayley graphs on $S_n$ that are line graphs, and classify all normal Cayley graphs on $S_n$ with the strictly second largest eigenvalue at most $1$.

Given a permutation $\sigma$, its corresponding inversion graph is obtained by adding an edge between $i<j$ if and only if $\sigma(i)>\sigma(j)$. The first results on random inversion graphs come from Acan and Pittel, who studied the connected threshold for a uniform permutation with fixed inversion number, and Bhattacharya and Mukherjee, who mostly focused on the degrees of the graph when the permutation is chosen uniformly at random. In this work, we call minimal inversion the minimal degree of the inversion graph and extend a theorem from Bhattacharya and Mukherjee to the case where the permutation is not only uniform, but obtained as the ordering of points sampled according to some distribution on the plane. Under regularity assumptions on the distribution, and for the appropriate $\alpha>0$, we show that the probability that the minimal inversion rescaled by $n^{\alpha/(\alpha+1)}$ is larger than $t$ behaves like $\exp(-ct^{\alpha+1})$ for some constant $c>0$ depending on the distribution. We further show that every $\alpha>0$ admits at least one corresponding distribution, thus proving that the minimal inversion can asymptotically scale as $n^\beta$ for any $\beta\in[0,1]$ (the cases $\beta=0$ and $\beta=1$ being obtained via the identity and anti-identity permutations, among others).

Let $\mathcal{K}_n$ be the simplex of nonnegative $n \times n$ matrices whose entries sum to $n$. For $A$ in $\mathcal{K}_n$, with row sums $r_1,\ldots,r_n$ and column sums $c_1,\ldots,c_n$, define $\Phi(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\mathrm{per}(A)$. Dittert conjectured that $\Phi$ is uniquely maximized by the uniform matrix $U_n=J_n/n$, with maximum $2-n!/n^n$. We prove the conjecture for every $n \geq 2$. Hwang's theorem already determines every positive maximizer, so the essential problem is to exclude maximizers on the boundary of $\mathcal{K}_n$. A reduction of Cheon and Wanless supplies two zero entries in distinct rows and columns. In order four, we classify all admissible support strata and exclude them by exact Karush-Kuhn-Tucker identities together with a Bernstein-form positivity certificate. For all $n \geq 5$, an attained-stratum averaging argument compresses a hypothetical boundary maximizer to a seven-parameter matrix with one multiplicity $m=n-2$. A symbolic odd-variable expansion, a cofactor identity at the two missing entries, and uniform coefficient inequalities then contradict the one-sided multiplier conditions. This argument is uniform in $m$ and uses no finite enumeration. We also give independent spectral proofs in dimensions 7 through 13 and prescribed-zero proofs in dimensions 14 and 15. Every dimension-dependent calculation is certified over the rationals and recorded explicitly in the appendices.

Let $N=2^k-1$ and let $\operatorname{wt}(n)$ denote the binary Hamming weight. The Tu-Deng conjecture asserts that, for every $1\le t\le N-1$, at most $2^{k-1}$ pairs $(a,b)\in\{0,\ldots,N-1\}^2$ satisfy $a+b\equiv t\pmod N$ and $\operatorname{wt}(a)+\operatorname{wt}(b)<k$. Partial results are known. We give a complete proof of this conjecture. We first show that the Tu-Deng counts equals the number of cyclic carry solutions for which $\operatorname{wt}(B)-\operatorname{wt}(A)<0$ and $A+t\equiv B\pmod N$. The enumerator of the cyclic carry solutions factors as $$C_v = 1+(X+Y-1)J_v+X^{\operatorname{z}(v)+1}Y^{\operatorname{o}(v)+1},$$ where $t=10v$ is the binary expansion of $t$(least significant bits first) and $J_v$ enumerates the language $$\operatorname{Sub}(v)\mathbin{\dot\cup}\{u\in\partial_1\operatorname{Sub}(v):u<_{\rm lex}v\}.$$ Estimating the strict negative half-plane mass of $C_v$ gives the desired bound.

We present three results on the multiset dimension of graphs, resolving one conjecture and two open questions from the literature. First, we disprove the conjecture of Simanjuntak, Siagian and Vetrík (2017) that every graph $G$ of order $n(G)$ with finite multiset dimension satisfies $\dim_m(G) \le n(G)-1$: an exhaustive computation over all 1,018,690,328 connected graphs of orders 2 through 11 shows that exactly eight graphs attain $\dim_m(G)=n(G)$, all of order 11, so 11 is the smallest order at which the trivial upper bound is attained. This also answers a question from the recent survey of Farhan, Klavžar, Kuziak and Yero. Second, we prove that $\dim_m(P_n \boxtimes P_n)=4$ for every $n \ge 5$, answering a question of Hakanen and Yero: after a $45^\circ$ change of coordinates the Chebyshev metric of the king grid becomes half the Manhattan metric on a parity sublattice, and four boundary inequalities reduce every potentially resolving three-landmark set to two geometric cases, in each of which we exhibit an explicit collision. Third, on king strips the parameter grows linearly: $\dim_m(P_3 \boxtimes P_n)=n$ for $n \ge 6$ (with the small cases determined exactly), where the lower bound rests on three local separation conditions and a finite min-plus transfer certificate whose equality case yields a finite automaton with a 19-state recurrent core, and the upper bound is an explicit landmark pattern of period three that works for every height. Combined with a blindness lower bound, $\dim_m(P_h \boxtimes P_n) = \Theta(n)$ for every fixed $h \ge 3$, so the constant answer on square king grids requires both dimensions to grow.

We prove that the dimer constant of the cubic lattice is bounded above by $0.452130$, improving the previous best upper bound of $0.457547$ obtained by Lundow (2001) using the transfer-matrix method of Ciucu (1998). Our construction instead uses diagonal transfer layers and an inequality of Csikvári (2017) to relate the spectral radius of the resulting matrix to the dimer constant, rather than exploiting symmetry special to the rectangular torus. We reduce the dimension of the transfer matrix using a compression method similar to Lundow's. Finally, we use a variant of the Collatz–Wielandt formula, derived from an asymptotic result of Friedland–Schneider (1980), to obtain a monotone sequence of upper bounds converging to the true spectral radius, each of which can be computed without forming the full matrix.

We provide a complete enumeration of the finite polyhedra that are noble, that is, polyhedra that are vertex transitive and facet transitive, through a computer-assisted proof which relates the possible existence of noble polyhedra to the roots of certain univariate or bivariate cubic polynomials. This is done through a parametrization of the orbits of point groups in $\mathbb{R}^3$, which naturally leads to a notion of criticality for certain orbits of a given point group, and a related equivalence relation $\equiv_c$ on the set of orbits. We show that within a given equivalence class of orbits under this relation, noble facetings are equivalent, and furthermore that there are a finite number of such equivalence classes. This gives a finite number of test cases, which may be tested through computer search. In total, we find that there are exactly 146 noble polyhedra in addition to the previously known infinite sets of stephanoids (crown polyhedra) and disphenoids.

For every $b\ge1$, let $Q_{2,b}$ be the ordered graph obtained from a transitive ordered triangle by attaching a monotone tail of length $b$ at its rightmost vertex. We prove $\rho_{<}(Q_{2,b})=\frac12$ for all $b\ge1$. Thus the previously isolated case $Q_{2,2}$ is one member of an exact infinite triangle-tail family. The lower bound is the sharp forward-template identity $\lambda(Q_{2,b})=1/2$, and it is realized already inside binary-level hosts: a parity-cut construction gives $Q_{2,b}$-free binary-level graphs with density exactly $1/2$ on every level. The upper bound uses the binary rich-level reduction. Its key input is the intrinsic decomposition of a $Q_{2,b}$-free graph into tail-starting vertices $T_b$ and the complement $R$: there are no forward edges from $R$ to $T_b$, $G[T_b]$ is ordered-triangle-free, and $G[R]$ is monotone-$\vec{P}_{b+1}$-free. We control these pieces by weighted binary $\vec{P}_{b+1}$ smoothing and weighted binary Mantel smoothing, the latter following from a binary ultrametric cut-domination theorem. We also record exact path-blow-up values, giving a reusable template/rich-host calculus for ordered relative densities.

We prove Schur-positivity for a family of symmetric functions.

For a family $\mathcal F\subseteq\binom{[n]}k$ and $R\in\binom{[n]}r$, let $d_{\mathcal F}(R)=|\{F\in\mathcal F:R\subseteq F\}|$ and $\ell_{r,p}(\mathcal F)=\sum_{R\in\binom{[n]}r}d_{\mathcal F}(R)^p$; write $co_p(\mathcal F)=\ell_{k-1,p}(\mathcal F)$ for the codegree power sum. We introduce a method that uses convexity to extend sharp bounds for degree sums and sums of squared degrees to real powers, while retaining all equality cases. The method bounds $x^p$ by quadratic polynomials or by a continuous function that is linear on each of two intervals. These functions agree with $x^p$ at the degrees of the proposed extremal family, so the argument requires no bounds for sums of higher powers. For families with bounded matching number, we instead use a bound for $\sum_R\max\{d_{\mathcal F}(R)-s,0\}$ together with the degree sum. We give three exact applications. First, a full $t$-star maximizes $co_p$ among $t$-intersecting families for every real $p\geq2$ in the sharp range $n\geq(t+1)(k-t+1)$, with all equality cases determined. This extends the quadratic theorem of Wu and Zhang to real exponents and answers a problem of Zhou and Yuan throughout the sharp Erdős–Ko–Rado range. Second, among intersecting families with $n\geq2k$, a full star maximizes $\ell_{r,p}$ for every $1\leq r\leq k-1$ and real $p\geq2$, again with all equality cases determined. Third, if $\nu(\mathcal F)\leq s$ and $n\geq(2s+1)k-s$, then for every real $p\geq1$, $co_p(\mathcal F)$ is uniquely maximized, up to permutation, by all $k$-sets meeting a fixed $s$-set. This removes the integrality restriction on $p$ and replaces previous cubic thresholds or assumptions that $n$ is sufficiently large with an explicit linear range valid for arbitrary $k$.

In this article, we present an explicit family of invertible $n\times n$ matrices over $\mathbb Z_2$ whose CNOT and row complexity is at least $4n-\text{o}(n)$; equivalently, reducing these matrices to the identity requires at least $4n-\text{o}(n)$ elementary row operations. Moreover, the same complexity lower bound holds in the stronger computational model where the CNOT gates are replaced by arbitrary local linear logic gates, namely arbitrary invertible linear transformations acting on pairs of coordinates. Let $G_n$ denote the permutation group generated by local logic gates acting on the set of binary strings of length $n$. We prove that $G_n$ is naturally isomorphic to the group of all invertible affine transformations of the vector space $\mathbb Z_2^n$, thus reducing the problem of estimating the quantum complexity of permutations in $G_n$ to the row reduction complexity of invertible matrices over $\mathbb Z_2$. As an application, we show that the permutations associated with our explicit matrices have quantum complexity at least $4n-\text{o}(n)$.

The Kemeny rule aggregates rankings by minimizing their total Kendall-tau distance from an aggregate order. We prove that Kemeny Score is NP-complete for exactly three unweighted rankings, even when every candidate pair is split $2$-to-$1$. On the same profiles, the winner, unique-winner, and possible- and necessary-precedence problems are $\Theta_2^p$-complete, while recognizing a Kemeny-optimal or uniquely Kemeny-optimal aggregate is coNP-complete. The hard instances induce tournaments of majority dimension exactly $3$. The reduction also determines the exact maximum-cut value from the optimal Kemeny score and recovers a maximum cut from any Kemeny-optimal aggregate. For every fixed $q\geq3$ and $\lceil q/2\rceil\leq s\leq q$, minimum pairwise support $s$ yields a sharp dichotomy: the score problem is NP-complete, the winner and precedence problems are $\Theta_2^p$-complete, and the recognition problems are coNP-complete when $3s\leq2q$; for $3s>2q$, the majority tournament is transitive and its unique topological order is the unique Kemeny-optimal aggregate. Exact support $s$ suffices in the hard case when $s>q/2$, and supports in ${s,s+1}$ suffice when $s=q/2$. These results give complete fixed-profile-size classifications and transfer to Slater orders, permutation medians, and maximum-likelihood central rankings in the Mallows model. Finally, a six-copy construction proves NP-completeness of both Kemeny Score and Kendall–Tau Center for three pairwise-equidistant rankings that still split every pair $2$-to-$1$. For $N$ output candidates, their common distance is $\frac23\binom N2$, the largest possible for an equidistant triple. The construction gives affine formulas for both optimal values, characterizes all Kemeny-optimal output orders, and shows that the output has a unique Kemeny-optimal order and a unique center exactly when the input has a unique Kemeny-optimal order.

In this note, using a result of Bernardi, Holden and Sun, we give an explicit geometric relation between the perimeter of the peeling process along the percolation interface of a triangulation, and the corresponding Kreweras walk. The relation naturally extends to the branching peeling exploration. This sheds light on the relation between the growth-fragmentation process and correlated Brownian excursions discovered by Da Silva, Powell and Watson.

Let $q=p^f$, and let $A\leq\mathbb{F}_q^\times$ be a multiplicative subgroup with $\mathbb{F}_p(A)=\mathbb{F}_q$. We prove that a proper subgroup $A$ is a generalized arithmetic progression (GAP) if and only if $|A| \in \{1, 2, 4\}$, and we determine when the full group $\mathbb{F}_q^\times$ is a GAP. For certain families of subgroups, we obtain the stronger conclusion that $A$ is additively irreducible. In particular, if $|A|>4$ and $p^e\equiv-1\pmod{|A|}$ for some $e\ge1$, then $A$ admits no nontrivial sumset decomposition. We also prove that every $c \neq 0$ has fewer than $|A|/2$ representations as a sum (or difference) of two elements of $A$ whenever $[\mathbb{F}_q^\times:A] \ge3$ and $|A| \ge 5$, which may be of independent interest.

We prove the redundancy of a family (the so-called melonic family) of large-$N$ moments in arbitrary tensor models. This reduces the number of independent entries in the positive semidefinite matrices that build the core of the tensor bootstrap (for tensor models there are already several generalizations of the positive semidefinite Toeplitz and Hankel matrices of lattice gauge theory and random matrix bootstrap, respectively). More concretely, we prove that any melonic operator $C$ at large-$N$ has an expectation value that depends on $C$ exclusively through its degree. A similar statement is shown here to hold for the Schwinger-Dyson equations of melonic moments. The strength of our result is also its scope, which is not limited to melonic tensor models.

For an $n$-dimensional convex body $K$, let $\theta_L(K)$ denote its lattice covering density, and let $\Theta_L^{\mathrm{conv}}(n)$ and $\Theta_L^{\mathrm{sym}}(n)$ be the corresponding worst-case quantities over all convex bodies and over origin-symmetric convex bodies, respectively. Before this work, these quantities were known only to lie between a lower bound of order $n$ and an upper bound of order $n^2$, so even their polynomial order was undetermined. We prove that there are absolute constants $c,C>0$ such that \[ c n\log n \le \Theta_L^{\mathrm{sym}}(n) \le \Theta_L^{\mathrm{conv}}(n) \le Cn\log n\,(\log\log n)^{10/3+o(1)}. \] Thus both worst-case quantities are $n\log n\,(\log n)^{o(1)}$, and the upper and lower bounds differ by a factor at most $(\log\log n)^{10/3+o(1)}$. For the upper bound, a vertical–horizontal amplification based on weighted Boolean cubes combines covering estimates for low-codimensional sections into an exact lattice covering of an arbitrary convex body. For the lower bound, a random-slab construction and Poisson witnesses on flat tori show, with positive probability, that the resulting body admits no lattice covering of density below $c n\log n$.

We prove that every positive rational number has eventually greedy best Egyptian underapproximations, both when repetitions of the denominators are allowed and when the denominators are required to be distinct. This answers affirmatively a problem originating with Erdős and Graham and later revisited by Nathanson, and yields an application concerning the maximal asymptotic growth of denominators in unit fraction series converging to a given rational number. We reformulate the question as an optimal control problem for a dynamical system, construct an appropriate payoff function, and study properties of the associated Bellman function. We also answer another question of Nathanson by constructing an irrational number with unique and greedy best Egyptian underapproximations.

Constructing log canonical compactifications of open varieties is a central problem in birational geometry. Finding a natural coordinate system and obtaining the equations of these models is difficult in general. We show that for a large class of varieties explicit coordinates for the log canonical model are provided by canonical forms of positive geometries, and use this to compute the equations of these models. Our theory applies, for instance, to complements of hyperplane arrangements, cubic surfaces with lines removed, and the moduli space of marked cubic del Pezzo surfaces.