In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest in this question dates back to the early 1980 s, when P. J. Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable

We prove that, for every graph F with at least one edge, there is a constant \(c_F\) such that there are graphs of arbitrarily large chromatic number and the same clique number as F in which every F-free induced subgraph has chromatic number at most \(c_F\). This generalises recent theorems of Briański, Davies and Walczak, and Carbonero, Hompe, Moore and Spirkl. Our results imply that for every \(r\geqslant

We study special cycles on the basic locus of certain unitary Shimura varieties over the ramified primes and their local analogs on the corresponding Rapoport–Zink spaces. We study the support and compute the dimension of these cycles.

We prove hybrid subconvexity bounds for GL ⁡ (3) × GL ⁡ (2) L-functions twisted by a primitive Dirichlet character modulo M (prime) in the M- and t-aspects. We also improve hybrid subconvexity bounds for twists of GL ⁡ (3) L-functions in the M- and t-aspects.

We give a computable lower bound for the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height bounds on components of the Noether–Lefschetz loci. This makes it possible to study the Diophantine properties of periods of quartic surfaces and to certify a part of the numerical computation of their Picard groups

The ring R of real-exponent polynomials in n variables over any field has global dimension n + 1 and flat dimension n. In particular, the residue field k ⁡ = R∕𝔪 of R modulo its maximal graded ideal 𝔪 has flat dimension n via a Koszul-like resolution. Projective and flat resolutions of all R-modules are constructed from this resolution of k ⁡ . The same results hold when R is replaced by the monoid

We develop machinery to explicitly determine, in many instances, when the difference x2 − yn is divisible only by powers of a given fixed prime. This combines a wide variety of techniques from Diophantine approximation (bounds for linear forms in logarithms, both archimedean and nonarchimedean, lattice basis reduction, methods for solving Thue–Mahler and S-unit equations, and the primitive divisor

We introduce a new class of Hodge cycles with nonreduced associated Hodge loci; we call them fake linear cycles. We characterize them for all Fermat varieties and show that they exist only for degrees d = 3,4,6, where there are infinitely many in the space of Hodge cycles. These cycles are pathological in the sense that the Zariski tangent space of their associated Hodge locus is of maximal dimension

Let \(A \subset {\mathbb {Z}}^d\) be a finite set. It is known that NA has a particular size (\(\vert NA\vert = P_A(N)\) for some \(P_A(X) \in {\mathbb {Q}}[X]\)) and structure (all of the lattice points in a cone other than certain exceptional sets), once N is larger than some threshold. In this article we give the first effective upper bounds for this threshold for arbitrary A. Such explicit results

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a \(P_5\)-free graph with clique number \(\omega \ge 3\) has chromatic number at most \(\omega ^{\log _2(\omega )}\). The best previous result was an exponential upper bound \((5/27)3^{\omega }\), due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdős-Hajnal conjecture holds

Let F be a nonarchimedean local field and G the F-points of a connected simply connected reductive group over F. We study the unipotent ℓ-blocks of G, for ℓ≠p. To that end, we introduce the notion of (d,1)-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and d-Harish-Chandra theory. The ℓ-blocks are then

We prove a characteristic p version of a theorem of Silverman on integral points in orbits over number fields and establish a primitive prime divisor theorem for polynomials in this setting. In characteristic p, the Thue–Siegel–Dyson–Roth theorem is false, so the proof requires new techniques from those used by Silverman. The problem is largely that isotriviality can arise in subtle ways, and we define

We classify additive operations in connective K-theory with various torsion-free coefficients. We discover that the answer for the integral case requires understanding of the ℤ^ case. Moreover, although integral additive operations are topologically generated by Adams operations, these are not reduced to infinite linear combinations of the latter ones. We describe a topological basis for stable operations

We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristic. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, which is a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with its twisted forms. The result actually

Let X ⊂ ℙn+e be any n-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property Nd,p(d ≥ 2,p ≥ 1), which means that X is d-regular up to p-th step in the minimal free resolution and the other is a new notion ND ⁡ (ℓ) which generalizes the classical “being nondegenerate” to the condition that requires a general finite linear section not to be contained

We prove a spectral reciprocity formula for automorphic forms on GL ⁡ (2) over a number field that is reminiscent of one found by Blomer and Khan. Our approach uses period representations of L-functions and the language of automorphic representations.

We prove that a smooth complete intersection of two quadrics of dimension at least 2 over a number field has index dividing 2, i.e., that it possesses a rational 0-cycle of degree 2.

We establish a p-adic Simpson correspondence in the spirit of Liu and Zhu for rigid analytic varieties X over ℂp with a liftable good reduction by constructing a new period sheaf on X proét. To do so, we use the theory of cotangent complexes described by Beilinson and Bhatt. Then we give an integral decompletion theorem and complete the proof by local calculations. Our construction is compatible with

Let G be a connected algebraic group and let X be a smooth projective G-variety. We prove a sufficient criterion to determine the bigness of the tangent bundle TX using the moment map ΦXG : T∗X → 𝔤∗. As an application, the bigness of the tangent bundles of certain quasihomogeneous varieties are verified, including symmetric varieties, horospherical varieties and equivariant compactifications of commutative

A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank 1 perturbation. Considered in this review are the additive rank 1 perturbation of the Hermitian Gaussian ensembles, the multiplicative rank 1 perturbation of the Wishart ensembles, and rank 1 perturbations of Hermitian and unitary matrices giving rise

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function \(f:\mathbb {N}\rightarrow \mathbb {N}\cup \{\infty \}\) with \(f(1)=1\) and \(f(n)\geqslant \left( {\begin{array}{c}3n+1\\ 3\end{array}}\right) \), we construct a hereditary class of graphs \({\mathcal {G}}\) such that the maximum chromatic number of a graph in \({\mathcal {G}}\) with clique number

We study families \({\mathcal {F}}\subseteq 2^{[n]}\) with restricted intersections and prove a conjecture of Snevily in a stronger form for large n. We also obtain stability results for Kleitman’s isodiametric inequality and families with bounded set-wise differences. Our proofs introduce a new twist to the classical linear algebra method, harnessing the non-shadows of \({\mathcal {F}}\), which may

Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weakly H-saturated in F if the edges in \(E(F)\setminus E(G)\) admit an ordering \(e_1,\ldots , e_k\) so that for all \(i\in [k]\) the hypergraph \(G\cup \{e_1,\ldots ,e_i\}\) contains an isomorphic copy of H which in turn contains the edge \(e_i\). The weak saturation number of H in F is the smallest

Consider two independent Poisson point processes of unit intensity in the Euclidean space of dimension d at least 3. We construct a perfect matching between the two point sets that is a factor (i.e., a measurable function of the point configurations that commutes with translations), and with the property that the distance between two matched configuration points has a tail distribution that decays

The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with K communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the

Motivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018), doi:10.1017/9781108231596 of Theorem 11.1.5], we show that this property holds for the ℓp-norm with 1≤p<∞ and i.i.d. ensemble

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if \(\log r/\log s\) is irrational and X and Y are \(\times r\) - and \(\times s\)-invariant subsets of [0, 1], respectively, then \(\dim _{\text {H}}(X + Y ) = \min (1, \dim _{\text {H}}X + \dim _{\text {H}}Y )\). Our main result yields information on the size of the sumset \(\lambda X

A family \({\mathcal {F}}\) on ground set \([n]:=\{1,2,\ldots , n\}\) is maximal k-wise intersecting if every collection of at most k sets in \({\mathcal {F}}\) has non-empty intersection, and no other set can be added to \({\mathcal {F}}\) while maintaining this property. In 1974, Erdős and Kleitman asked for the minimum size of a maximal k-wise intersecting family. We answer their question for \(k=3\)

Linear sets in projective spaces over finite fields were introduced by Lunardon (Geom Dedic 75(3):245–261, 1999) and they play a central role in the study of blocking sets, semifields, rank-metric codes, etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line

Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics (LSS) via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type

We survey the area of strongly regular graphs satisfying the 4-vertex condition and find several new families. We describe a switching operation on collinearity graphs of polar spaces that produces cospectral graphs. The obtained graphs satisfy the 4-vertex condition if the original graph belongs to a symplectic polar space.

The main results of this paper concern growth in sums of a k-convex function f. Firstly, we streamline the proof (from Hanson et al. (Combinatorica 42:71–85, 2020)) of a growth result for f(A) where A has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for $$\begin{aligned} |2^k f(A) - (2^k-1)f(A)|. \end{aligned}$$ We also generalise

Katona’s intersection theorem states that every intersecting family \(\mathcal {F}\subseteq [n]^{(k)}\) satisfies \(\vert \partial \mathcal {F}\vert \geqslant \vert \mathcal {F}\vert \), where \(\partial \mathcal {F}=\{F\setminus \{x\}:x\in F\in \mathcal {F}\}\) is the shadow of \(\mathcal {F}\). Frankl conjectured that for \(n>2k\) and every intersecting family \(\mathcal {F}\subseteq [n]^{(k)}\)

We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window.

Let [X,λ] be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either X is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor νv([X,λ]) for each place v of ℚ, and show that the product of these factors essentially computes the size of the isogeny class of [X,λ]. The derivation of this mass

Let V,W be a pair of smooth varieties. We want to compare curve counts on V × W with those on V and W. The product formula in Gromov–Witten theory compares the virtual fundamental classes of stable maps to a product M¯g,n(V × W) to the product of stable maps M¯g,n(V ) ×M¯g,n(W). We prove the analogous theorem for log stable maps to log smooth varieties V,W. This extends results of Y.P. Lee and F. Qu

We introduce the modular intersection kernel, and we use it to study how geodesics intersect on the full modular surface 𝕏 = PSL ⁡ 2(ℤ)∖ℍ. Let Cd be the union of closed geodesics with discriminant d and let β ⊂ 𝕏 be a compact geodesic segment. As an application of Duke’s theorem to the modular intersection kernel, we prove that {(p,𝜃p) : p ∈ β ∩ Cd} becomes equidistributed with respect to sin ⁡

We prove that the number of irreducible ordinary characters in the principal p-block of a finite group G of order divisible by p is always at least 2p − 1. This confirms a conjecture of Héthelyi and Külshammer (2000) for principal blocks and provides an affirmative answer to Brauer’s problem 21 (1963) for principal blocks of bounded defect. Our proof relies on recent works of Maróti (2016) and Malle

We construct a new Eisenstein cocycle, called the Shintani–Barnes cocycle, which specializes in a uniform way to the values of the zeta functions of general number fields at positive integers. Our basic strategy is to generalize the construction of the Eisenstein cocycle presented in the work of Vlasenko and Zagier by using some recent techniques developed by Bannai, Hagihara, Yamada, and Yamamoto

Let G be a finite group, let p be a prime and let Pr ⁡ p(G) be the probability that two random p-elements of G commute. In this paper we prove that Pr ⁡ p(G) > (p2 + p − 1)∕p3 if and only if G has a normal and abelian Sylow p-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime p there

This is a correction to the paper “Height bounds and the Siegel property” (Algebra Number Theory 12:2 (2018), 455–478). We correct an error in the proof of Theorem 4.1. Theorem 4.1 as stated in the original paper is correct, but the correction affects additional information about the theorem which is important for applications.