@article{22163, abstract = {For a field F and integers d and k, a set A ⊆ Fd is called k-nearly orthogonal if its members are non-self-orthogonal and every k + 1 vectors of A include an orthogonal pair. We prove that for every prime p there exists some δ = δ(p)> 0, such that for every field F of characteristic p and for all integers k ≥ 2 and d ≥ k, there exists a k-nearly orthogonal set of at least dδ·k/ logk vectors of Fd. The size of the set is optimal up to the logk term in the exponent. We further prove two extensions of this result. In the first, we provide a large set A of non-self-orthogonal vectors of Fd such that for every two subsets of A of size k+1 each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every k + 1 vectors of the produced set A include ℓ + 1 pairwise orthogonal vectors for an arbitrary fixed integer 1 ≤ ℓ ≤ k. The proofs involve probabilistic and spectral arguments and the hypergraph container method}, author = {Haviv, Ishay and Mattheus, Sam and Milojević, Aleksa and Wigderson, Yuval}, issn = {0012-365X}, journal = {Discrete Mathematics}, keywords = {Nearly orthogonal sets, Ramsey theory, Finite fields}, number = {4}, publisher = {Elsevier}, title = {{Larger nearly orthogonal sets over finite fields}}, doi = {10.1016/j.disc.2024.114373}, volume = {348}, year = {2025}, } @article{22168, abstract = {Let us say that a graph G is Ramsey for a tuple (H1, ... , Hr) of graphs if every r-colouring of the edges of G contains a monochromatic copy of Hi in colour i, for some i ∈ [[r]]. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and Rucinski, predicts the threshold at which the binomial random graph ´ Gn,p becomes Ramsey for (H1, ... , Hr) asymptotically almost surely. In this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement, which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs H1, ... , Hr by finite families H1, ... , Hr. Additionally, we pose a natural (deterministic) graph-partitioning conjecture, which we believe to be of independent interest, and whose resolution would imply the Kohayakawa–Kreuter conjecture.}, author = {KUPERWASSER, EDEN and SAMOTIJ, WOJCIECH and Wigderson, Yuval}, issn = {1469-8064}, journal = {Mathematical Proceedings of the Cambridge Philosophical Society}, number = {3}, pages = {293--320}, publisher = {Cambridge University Press}, title = {{On the Kohayakawa–Kreuter conjecture}}, doi = {10.1017/s0305004125000143}, volume = {178}, year = {2025}, } @article{22172, abstract = {A highly influential result of Nikiforov states that if an n-vertex graph G contains at least γnh copies of a fixed h-vertex graph H, then G contains a blowup of H of order Ωγ,H(logn). While the dependence on n is optimal, the correct dependence on γ is unknown; all known proofs yield bounds that are polynomial in γ, but the best known upper bound, coming from random graphs, is only logarithmic in γ. It is a major open problem to narrow this gap. We prove that if H is triangle-free, then the logarithmic behavior of the upper bound is the truth. That is, under the assumptions above, G contains a blowup of H of order ΩH(logn/log(1/γ)). This is the first non-trivial instance where the optimal dependence in Nikiforov’s theorem is known. As a consequence, we also prove an upper bound on multicolor Ramsey numbers of blowups of triangle-free graphs, proving that the dependence on the number of colors is polynomial once the blowup is sufficiently large. This shows that, from the perspective of multicolor Ramsey numbers, blowups of fixed triangle-free graphs behave like bipartite graphs.}, author = {Girão, António and Hunter, Zach and Wigderson, Yuval}, journal = {Advances in Combinatorics}, publisher = {Alliance of Diamond Open Access Journals}, title = {{Blowups of triangle-free graphs}}, doi = {10.19086/aic.2025.10}, volume = {10}, year = {2025}, } @article{22181, abstract = {A graph G is said to be Ramsey size-linear if r(G, H) = OG(e(H)) for every graph H with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that K4 is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative}, author = {Wigderson, Yuval}, issn = {0195-6698}, journal = {European Journal of Combinatorics}, publisher = {Elsevier}, title = {{Infinitely many minimally non-Ramsey size-linear graphs}}, doi = {10.1016/j.ejc.2025.104175}, volume = {128}, year = {2025}, } @article{22188, abstract = {A fundamental fact about bounded-degree graph expanders is that three notions of expansion—vertex expansion, edge expansion, and spectral expansion—are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely, dimension expansion (defined in analogy to vertex expansion of graphs) and quantum expansion (defined in analogy to spectral expansion of graphs). Lubotzky and Zelmanov proved that the latter implies the former. We prove that the converse is false: There are dimension expanders which are not quantum expanders. This also answers in the negative questions of Lubotzky--Zelmanov and Dvir--Shpilka on the relation between dimension expansion and Kazhdan's property T. Moreover, this asymmetry is explained by the fact that there are two distinct linear-algebraic analogues of edge expansion of graphs. The first of these is quantum edge expansion, which was introduced by Hastings, and which he proved to be equivalent to quantum expansion. We introduce a new notion, termed dimension edge expansion, which we prove is equivalent to dimension expansion and which is implied by quantum edge expansion. Thus, the separation above is implied by a finer one: dimension edge expansion is strictly weaker than quantum edge expansion. This new notion also leads to a new, more modular proof of the Lubotzky--Zelmanov result that quantum expanders are dimension expanders.}, author = {Li, Yinan and Qiao, Youming and Wigderson, Avi and Wigderson, Yuval and Zhang, Chuanqi}, issn = {1557-2862}, journal = {Theory of Computing}, keywords = {linear algebraic expansion, quantum expanders, dimension expanders}, publisher = {Theory of Computing Exchange}, title = {{On linear-algebraic notions of expansion}}, doi = {10.4086/toc.2025.v021a001}, volume = {21}, year = {2025}, } @article{22191, abstract = {Montgomery and Soundararajan showed that the distribution of ψ(x+H)−ψ(x), for 0≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log(N/H), when N δ ≤ H ≤ N 1−δ . Their work depends on showing that sums Rk (h) of k-term singular series are µk (−h log h + Ah) k/2 + Ok (h k/2−1/(7k)+ε ), where A is a constant and µk are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when k is odd, Rk (h) ≍ h (k−1)/2 (log h) (k+1)/2 . We prove an upper bound with the correct power of h when k = 3, and prove analogous upper bounds in the function field setting when k = 3 and k = 5. We provide further evidence for this conjecture in the form of numerical computations.}, author = {Kuperberg, Vivian Zieve}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {4}, pages = {617--666}, publisher = {Mathematical Sciences Publishers}, title = {{Odd moments in the distribution of primes}}, doi = {10.2140/ant.2025.19.617}, volume = {19}, year = {2025}, } @article{22195, abstract = {Sums of the singular series constants that appear in the Hardy–Littlewood k-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo r or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo r of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats. }, author = {Kuperberg, Vivian Zieve}, issn = {1793-7310}, journal = {International Journal of Number Theory}, number = {01}, pages = {53--74}, publisher = {World Scientific Publishing}, title = {{Sums of singular series along arithmetic progressions and with smooth weights}}, doi = {10.1142/s1793042125500046}, volume = {21}, year = {2025}, } @article{22200, abstract = {In Kuperberg and Lal´ın [Forum Math. 34 (2022), pp. 711–747], the authors studied the mean-square of certain sums of the divisor function dk(f) over the function field Fq[T] in the limit as q →∞ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [Math. Z. 288 (2018), pp. 167–198] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields.}, author = {Kuperberg, Vivian Zieve and Lalín, Matilde}, issn = {2330-0000}, journal = {Transactions of the American Mathematical Society, Series B}, number = {10}, pages = {323--370}, publisher = {American Mathematical Society}, title = {{Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime}}, doi = {10.1090/btran/186}, volume = {12}, year = {2025}, } @article{22199, abstract = {Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function dk(n) over number fields, which were inspired by analogous results over function fields proven by the authors. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.}, author = {Kuperberg, Vivian Zieve and Lalín, Matilde}, issn = {0025-5793}, journal = {Mathematika}, number = {3}, publisher = {Wiley}, title = {{Arithmetic constants for symplectic variances of the divisor function}}, doi = {10.1112/mtk.70029}, volume = {71}, year = {2025}, } @article{22201, abstract = {We consider an analog of a conjecture of Montgomery and Soundararajan on the moments of primes in short intervals in number fields; this analog was discussed and heuristically derived in a paper of the second author, Rodgers, and Roditty-Gershon. Adapting work of the first author and Fiorilli in the integer case, we establish lower bounds on a weighted version of these moments which agree with the conjectured values.}, author = {de la Bretèche, Régis and Kuperberg, Vivian Zieve}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, number = {1}, pages = {437--461}, publisher = {Springer Nature}, title = {{Lower bounds on weighted moments of primes in short intervals in number fields}}, doi = {10.1007/s11856-024-2711-0}, volume = {267}, year = {2025}, } @article{22203, abstract = {We prove near-optimal upper bounds for the oddmoments of the distribution of coprime residues inshort intervals, confirming a conjecture of Montgomeryand Vaughan. As an application, we prove near-optimalupper bounds for the average of the refined singularseries in the Hardy–Littlewood conjectures concerningthe number of prime 𝑘-tuples for 𝑘 odd. The mainnew ingredient is a near-optimal upper bound for thenumber of solutions to ∑1 ⩽𝑖 ⩽𝑘𝑎 𝑖𝑞𝑖∈ ℤ when 𝑘 is odd,with gcd(𝑎𝑖 , 𝑞𝑖 ) = 1 and restrictions on the size of thenumerators and denominators, which is of indepen-dent interest.}, author = {Bloom, Thomas F. and Kuperberg, Vivian Zieve}, issn = {1460-244X}, journal = {Proceedings of the London Mathematical Society}, number = {1}, publisher = {Wiley}, title = {{Odd moments and adding fractions}}, doi = {10.1112/plms.70068}, volume = {131}, year = {2025}, } @article{22204, abstract = {We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q, any two reduced congruence classes a1 and a2 mod q, and any r1,r2 ≥ 1, a positive density of sums of two squares begin a chain of r1 consecutive sums of two squares, all of which are a1 mod q, followed immediately by a chain of r2 consecutive sums of two squares, all of which are a2 mod q. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any r ≥ 1, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q}, author = {Kimmel, Noam and Kuperberg, Vivian Zieve}, issn = {1475-3030}, journal = {Journal of the Institute of Mathematics of Jussieu}, number = {5}, pages = {1995--2046}, publisher = {Cambridge University Press}, title = {{Positive density for consecutive runs of sums of two squares}}, doi = {10.1017/s1474748025000131}, volume = {24}, year = {2025}, } @article{22213, abstract = {Unlike biological active matter that constantly adapt to their environment, the motors of synthetic active particles are typically agnostic to their surroundings and merely operate at constant force. Here, we design colloidal active rods capable of modulating their inner activity in response to crowding, thereby enforcing a primitive form of quorum sensing interactions. Through experiments, simulations, and theory we elucidate the impact of these interactions on the phase behavior of isotropic active matter. We demonstrate that, when conditioned to density, motility regulation can either lead to an absorbing phase transition, where all particles freeze their dynamics, or to atypical phase separation, where flat interfaces supporting a net pressure drop are in mechanical equilibrium. Fully active and fully arrested particles can then form heterogeneous patterns ruled by the competition between quorum sensing and mechanical interactions. Beyond the specifics of motile colloids, we expect our findings to apply broadly to adaptive active matter assembled from living or synthetic units.}, author = {Lefranc, Thibault and Dinelli, Alberto and Fernández-Rico, Carla and Dullens, Roel P. A. and Tailleur, Julien and Bartolo, Denis}, issn = {2160-3308}, journal = {Physical Review X}, number = {3}, publisher = {American Physical Society}, title = {{Synthetic quorum sensing and absorbing phase transitions in colloidal active matter}}, doi = {10.1103/8csn-71jk}, volume = {15}, year = {2025}, } @article{22214, abstract = {Highly interconnected percolated networks are interesting structures for materials with enhanced transport and mechanical properties. While percolated networks of anisotropic particles have been explored at the nanoscale, achieving highly interconnected structures at the microscale remains challenging. In this work, we explore the controlled assembly of rod-like polymer colloids under external fields leading to reversible quasi-2D networks. By varying voltage and frequency, we modulate the pore size and thickness of the network. We find that field-driven attractive interactions enable percolation at lower area fractions than predicted for non-interacting rods. Monte Carlo simulations incorporating dipolar interactions and electrostatic boundary conditions confirm the field-induced transition from isotropic to aligned rod configurations, supporting the emergence of percolated networks. This work presents a simple and robust approach for assembling reconfigurable colloidal networks with controlled connectivity, offering new strategies for designing adaptive soft materials.}, author = {Fojo, José and Subert, Rodolfo and Rodríguez-Arco, Laura and López-López, Modesto T. and Dijkstra, Marjolein and Fernández-Rico, Carla and Alvarez, Laura}, issn = {1744-6848}, journal = {Soft Matter}, number = {23}, pages = {4596--4605}, publisher = {Royal Society of Chemistry}, title = {{Field-driven reversible networks from colloidal rods}}, doi = {10.1039/d5sm00218d}, volume = {21}, year = {2025}, } @article{12311, abstract = {In this note, we prove a formula for the cancellation exponent kv,n between division polynomials ψn and ϕn associated with a sequence {nP}n∈N of points on an elliptic curve E defined over a discrete valuation field K. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.}, author = {Naskręcki, Bartosz and Verzobio, Matteo}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics}, keywords = {Elliptic curves, Néron models, division polynomials, height functions, discrete valuation rings}, number = {5}, pages = {1646--1660}, publisher = {Cambridge University Press}, title = {{Common valuations of division polynomials}}, doi = {10.1017/prm.2024.7}, volume = {155}, year = {2025}, } @article{20797, abstract = {Quantum key distribution (QKD) offers a theoretically secure method to share secret keys, yet practical implementations face challenges due to noise and loss over long-distance channels. Traditional QKD protocols require extensive noise compensation, hindering their industrial scalability and lowering the achievable key rates. Alternative protocols encode logical qubits in noise-resilient states but at the cost of using many physical qubits, increasing susceptibility to loss and limiting transmission distance. In this work, we introduce a logical-qubit encoding that uses antisymmetric Bell states in the continuous photonic degrees of freedom, frequency and time. By leveraging the continuous space, we overcome this noise-loss robustness trade-off by minimizing the number of photons per logical qubit while optimizing the encoding resilience over noise fluctuations. We analyze the security of our encoding and demonstrate its robustness compared to existing state-of-the-art protocols. This approach provides a path toward scalable, efficient QKD implementations under realistic noise conditions.}, author = {Seabrook, Hannah and Lavie, Emilien and Strömberg, Karl T and Stafford, Matthew P. and Rubino, Giulia}, issn = {2331-7019}, journal = {Physical Review Applied}, number = {2}, publisher = {American Physical Society}, title = {{Surpassing the loss-noise robustness trade-off in quantum key distribution}}, doi = {10.1103/xq2l-r4r7}, volume = {24}, year = {2025}, } @article{21244, abstract = {Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds. }, author = {Browning, Timothy D and Lyczak, Julian and Smeets, Arne}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {10}, pages = {2049--2090}, publisher = {Mathematical Sciences Publishers}, title = {{Paucity of rational points on fibrations with multiple fibres}}, doi = {10.2140/ant.2025.19.2049}, volume = {19}, year = {2025}, } @article{20850, abstract = {We provide an estimate for the number of nontrivial integer points on the Pellian surface t^2 - du^2 = 1 in a bounded region. We give a lower bound on the size of fundamental solutions for almost all d in a certain class, based on a recent conjecture of Browning and Wilsch about integer points on log K3 surfaces. We also obtain an upper bound on the average of class number in this class, assuming the same conjecture.}, author = {Diao, Yijie}, issn = {2118-8572}, journal = {Journal de theorie des nombres de Bordeaux}, number = {3}, pages = {973--988}, publisher = {Université de Bordeaux}, title = {{Class numbers and integer points on some Pellian surfaces}}, doi = {10.5802/jtnb.1348}, volume = {37}, year = {2025}, } @article{21003, abstract = {We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.}, author = {Browning, Timothy D and Verzobio, Matteo}, issn = {2397-3129}, journal = {Discrete Analysis}, publisher = {Cambridge: Alliance of Diamond Open Access Journals}, title = {{Counting integer points on affine surfaces with a side condition}}, doi = {10.19086/da.143787}, volume = {2025}, year = {2025}, } @article{21260, abstract = {We prove that there does not exist F∈Q[x,y] of degree 4 such that F(Z^2 )=Z ≥0. In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.}, author = {Yao Xiao, Stanley and Yamagishi, Shuntaro}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {EMS Press}, title = {{Quartic polynomials in two variables do not represent all non-negative integers}}, doi = {10.4171/jems/1697}, year = {2025}, }