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Moments of zeta and correlations of divisor-sums: Stratification and Vandermonde integrals Mathematika (IF 0.8) Pub Date : 2024-03-15 Siegfred Baluyot, Brian Conrey
We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group
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Zeros of modular forms and Faber polynomials Mathematika (IF 0.8) Pub Date : 2024-03-13 Zeév Rudnick
We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near vertical lines, with the zeros of a weight form lying at height approximately . This is in contrast to previously known cases, such
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High moments of theta functions and character sums Mathematika (IF 0.8) Pub Date : 2024-02-14 Barnabás Szabó
Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet L-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen theorems of Munsch, who proved almost sharp upper bounds for these quantities. The main new ingredient of our proof comes from a paper of Harper, who showed the related
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Hausdorff dimension of Besicovitch sets of Cantor graphs Mathematika (IF 0.8) Pub Date : 2024-02-08 Iqra Altaf, Marianna Csörnyei, Kornélia Héra
We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets Γ, that is, sets that contain a rotated copy of Γ in each direction. We show that for a large class of Cantor sets C and Cantor-graphs Γ built on C, the Hausdorff dimension of any Γ-Besicovitch set must be at least , where .
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On the Lindelöf hypothesis for general sequences Mathematika (IF 0.8) Pub Date : 2024-02-05 Frederik Broucke, Sebastian Weishäupl
In a recent paper, Gonek, Graham, and Lee introduced a notion of the Lindelöf hypothesis (LH) for general sequences that coincides with the usual LH for the Riemann zeta function in the case of the sequence of positive integers. They made two conjectures: that LH should hold for every admissible sequence of positive integers, and that LH should hold for the “generic” admissible sequence of positive
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Zeros of dirichlet L-functions near the critical line Mathematika (IF 0.8) Pub Date : 2024-01-04 George Dickinson
We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet L-functions of modulus q at height T. To do this, we derive an asymptotic for the twisted second moment of Dirichlet L-functions uniformly in q and t. As a second application of the asymptotic formula, we prove that, for every integer q, at least 38.2% of zeros of the primitive Dirichlet L-functions
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Discrepancy of arithmetic progressions in grids Mathematika (IF 0.8) Pub Date : 2024-01-03 Jacob Fox, Max Wenqiang Xu, Yunkun Zhou
We prove that the discrepancy of arithmetic progressions in the d-dimensional grid { 1 , ⋯ , N } d $\lbrace 1, \dots, N\rbrace ^d$ is within a constant factor depending only on d of N d 2 d + 2 $N^{\frac{d}{2d+2}}$ . This extends the case d = 1 $d=1$ , which is a celebrated result of Roth and of Matoušek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valkó from
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A quantitative Hasse principle for weighted quartic forms Mathematika (IF 0.8) Pub Date : 2023-12-26 Daniel Flores
We derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials F ( x , y ) ∈ Z [ x 1 , … , x s 1 , y 1 , … , y s 2 ] $F({\mathbf {x}},{\mathbf {y}}) \in {\mathbb {Z}}[x_1,\ldots ,x_{s_1},y_1,\ldots ,y_{s_2}]$ satisfy the condition that F ( λ 2 x
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A unified approach to higher order discrete and smooth isoperimetric inequalities Mathematika (IF 0.8) Pub Date : 2023-12-20 Kwok-Kun Kwong
We present a unified approach to derive sharp isoperimetric-type inequalities of arbitrary high order. In particular, we obtain (i) sharp high-order discrete polygonal isoperimetric-type inequalities, (ii) sharp high-order isoperimetric-type inequalities for smooth curves with both upper and lower bounds for the isoperimetric deficit, and (iii) sharp higher order Chernoff-type inequalities involving
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On the homotopy type of multipath complexes Mathematika (IF 0.8) Pub Date : 2023-12-09 Luigi Caputi, Carlo Collari, Sabino Di Trani, Jason P. Smith
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph G ${\tt G}$ is the simplicial complex whose faces are the multipaths of G ${\tt G}$ . We compute Euler characteristics, and associated generating functions, of the multipath complexes of directed graphs from certain families, including transitive tournaments and complete bipartite graphs. We show
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Kloosterman sums do not correlate with periodic functions Mathematika (IF 0.8) Pub Date : 2023-11-19 Raphael S. Steiner
We provide uniform bounds for sums of Kloosterman sums in all arithmetic progressions. As a consequence, we find that Kloosterman sums do not correlate with periodic functions.
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A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem Mathematika (IF 0.8) Pub Date : 2023-11-12 Brian Cook, Kevin Hughes, Zane Kun Li, Akshat Mudgal, Olivier Robert, Po-Lam Yung
We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory.
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Quasi-invariance of modulus and quasisymmetry of weakly (L,M)-quasisymmetric maps in metric spaces Mathematika (IF 0.8) Pub Date : 2023-11-12 Tao Cheng, Shanshuang Yang
This paper contributes to the study of a fundamental problem in the theory of quasiconformal analysis: under what conditions local quasiconformality of a homeomorphism implies its global quasisymmetry. We show that in general metric spaces local regularity and some connectivity together with the Loewner condition are necessary and sufficient for a quasiconformal map to be globally quasisymmetric with
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Coloring unions of nearly disjoint hypergraph cliques Mathematika (IF 0.8) Pub Date : 2023-11-09 Dhruv Mubayi, Jacques Verstraete
We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are generally known to exist only when the number of cliques is exponential in the clique size (Glock, Kühn, Lo, and Osthus, Mem. Amer. Math. Soc. 284 (2023) v+131 pp; Keevash, Preprint; Rödl, Eur. J. Combin
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Correlation of multiplicative functions over Fq[x]$\mathbb {F}_q[x]$: A pretentious approach Mathematika (IF 0.8) Pub Date : 2023-10-26 Pranendu Darbar, Anirban Mukhopadhyay
Let M n $\mathcal {M}_n$ denote the set of monic polynomials of degree n over a finite field F q $\mathbb {F}_q$ of q elements. For multiplicative functions ψ 1 , ψ 2 $\psi _1,\psi _2$ , using the recently developed “pretentious method,” we establish a “local-global” principle for correlation functions of the form
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Intrinsic Diophantine approximation on circles and spheres Mathematika (IF 0.8) Pub Date : 2023-10-26 Byungchul Cha, Dong Han Kim
We study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in R 2 $\mathbb {R}^2$ or R 3 $\mathbb {R}^3$ and three spheres embedded in R 3 $\mathbb {R}^3$ or R 4 $\mathbb {R}^4$ . We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of R $\mathbb {R}$ and C $\mathbb
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On the abc$abc$ conjecture in algebraic number fields Mathematika (IF 0.8) Pub Date : 2023-10-26 Andrew Scoones
In this paper, we prove a weak form of the a b c $abc$ conjecture generalised to algebraic number fields. Given integers satisfying a + b = c $a+b=c$ , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential
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Upper bounds for the constants of Bennett's inequality and the Gale–Berlekamp switching game Mathematika (IF 0.8) Pub Date : 2023-10-27 Daniel Pellegrino, Anselmo Raposo
In 1977, G. Bennett proved, by means of nondeterministic methods, an inequality that plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for p 1 , p 2 ∈ [ 1 , ∞ ] $p_{1},p_{2} \in [1,\infty ]$ and all positive integers n 1 , n 2 $n_{1},n_{2}$ , there exists a bilinear form A n 1 , n 2 : ( R n 1 , ∥ · ∥ p 1 ) × ( R n 2 , ∥ · ∥ p 2 ) ⟶ R $A_{n_{1}
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Optimal Hardy-weights for elliptic operators with mixed boundary conditions Mathematika (IF 0.8) Pub Date : 2023-09-28 Yehuda Pinchover, Idan Versano
We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator ( P , B ) $(P,B)$ with degenerate mixed boundary conditions. By an optimal Hardy-weight for a subcritical operator we mean a nonzero nonnegative weight function W such that ( P − W , B ) $(P-W,B)$ is critical, and null-critical with respect to W. Our results rely on a recently developed criticality
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On the number of tiles visited by a line segment on a rectangular grid Mathematika (IF 0.8) Pub Date : 2023-09-30 Alex Arkhipov, Luis Mendo
Consider a line segment placed on a two-dimensional grid of rectangular tiles. This paper addresses the relationship between the length of the segment and the number of tiles it visits (i.e., has intersection with). The square grid is also considered explicitly, as some of the specific problems studied are more tractable in that particular case. The segment position and orientation can be modeled as
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Stability of polydisc slicing Mathematika (IF 0.8) Pub Date : 2023-09-27 Nathaniel Glover, Tomasz Tkocz, Katarzyna Wyczesany
We prove a dimension-free stability result for polydisc slicing due to Oleszkiewicz and Pełczyński. Intriguingly, compared to the real case, there is an additional asymptotic maximizer. In addition to Fourier-analytic bounds, we crucially rely on a self-improving feature of polydisc slicing, established via probabilistic arguments.
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On the distribution of modular inverses from short intervals Mathematika (IF 0.8) Pub Date : 2023-09-27 Moubariz Z. Garaev, Igor E. Shparlinski
For a prime number p and integer x with gcd ( x , p ) = 1 $\gcd (x,p)=1$ , let x ¯ $\overline{x}$ denote the multiplicative inverse of x modulo p. In this paper, we are interested in the problem of distribution modulo p of the sequence
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Two problems on the distribution of Carmichael's lambda function Mathematika (IF 0.8) Pub Date : 2023-09-27 Paul Pollack
Let λ ( n ) $\lambda (n)$ denote the exponent of the multiplicative group modulo n. We show that when q is odd, each coprime residue class modulo q is hit equally often by λ ( n ) $\lambda (n)$ as n varies. Under the stronger assumption that gcd ( q , 6 ) = 1 $\gcd (q,6)=1$ , we prove that equidistribution persists throughout a Siegel–Walfisz-type range of uniformity. By similar methods we show that
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A dichotomy phenomenon for bad minus normed Dirichlet Mathematika (IF 0.8) Pub Date : 2023-08-14 Dmitry Kleinbock, Anurag Rao
Given a norm ν on R 2 $\mathbb {R}^2$ , the set of ν-Dirichlet improvable numbers DI ν $\mathbf {DI}_\nu$ was defined and studied in the papers (Andersen and Duke, Acta Arith. 198 (2021) 37–75 and Kleinbock and Rao, Internat. Math. Res. Notices 2022 (2022) 5617–5657). When ν is the supremum norm, DI ν = BA ∪ Q $\mathbf {DI}_\nu = \mathbf {BA}\cup {\mathbb {Q}}$ , where BA $\mathbf {BA}$ is the set
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Families of ϕ-congruence subgroups of the modular group Mathematika (IF 0.8) Pub Date : 2023-08-13 Angelica Babei, Andrew Fiori, Cameron Franc
We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed ϕ-congruence subgroups, are obtained by reducing homomorphisms ϕ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the
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Lower bounds for negative moments of ζ′(ρ)$\zeta ^{\prime }(\rho )$ Mathematika (IF 0.8) Pub Date : 2023-08-10 Peng Gao, Liangyi Zhao
We establish lower bounds for the discrete 2kth moment of the derivative of the Riemann zeta function at nontrivial zeros for all k < 0 $k<0$ under the Riemann hypothesis and the assumption that all zeros of ζ ( s ) $\zeta (s)$ are simple.
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Averages of long Dirichlet polynomials with modular coefficients Mathematika (IF 0.8) Pub Date : 2023-08-09 Brian Conrey, Alessandro Fazzari
We study the moments of L-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a long Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindelöf Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.
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M0, 5: Toward the Chabauty–Kim method in higher dimensions Mathematika (IF 0.8) Pub Date : 2023-08-07 Ishai Dan-Cohen, David Jarossay
If Z is an open subscheme of Spec Z $\operatorname{Spec}\mathbb {Z}$ , X is a sufficiently nice Z-model of a smooth curve over Q $\mathbb {Q}$ , and p is a closed point of Z, the Chabauty–Kim method leads to the construction of locally analytic functions on X ( Z p ) $X({\mathbb {Z}_p})$ which vanish on X ( Z ) $X(Z)$ ; we call such functions “Kim functions”. At least in broad outline, the method generalizes
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Banach spaces of continuous functions without norming Markushevich bases Mathematika (IF 0.8) Pub Date : 2023-07-28 Tommaso Russo, Jacopo Somaglia
We investigate the question whether a scattered compact topological space K such that C ( K ) $C(K)$ has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that C ( [ 0 , ω 1 ] ) $C([0,\omega _1])$ does not
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On simply normal numbers with digit dependencies Mathematika (IF 0.8) Pub Date : 2023-07-14 Verónica Becher, Agustín Marchionna, Gérald Tenenbaum
Given an integer b ⩾ 2 $b\geqslant 2$ and a set P ${\EuScript P}$ of prime numbers, the set T P ${\EuScript T}_{\EuScript P}$ of Toeplitz numbers comprises all elements of [0, b[ whose digits ( a n ) n ⩾ 1 $(a_n)_{n\geqslant 1}$ in the base-b expansion satisfy a n = a p n $a_n=a_{pn}$ for all p ∈ P $p\in {\EuScript P}$ and n ⩾ 1 $n\geqslant 1$ . Using a completely additive arithmetical function, we
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The functional Orlicz–Brunn–Minkowski inequality for q-torsional rigidity Mathematika (IF 0.8) Pub Date : 2023-06-26 Jinrong Hu, Ping Zhang
In this paper, we obtain the functional Orlicz–Brunn–Minkowski inequality and the functional Orlicz–Minkowski inequality for q-torsional rigidity in the smooth category. Furthermore, using an approximation method, we give the general functional Orlicz–Brunn–Minkowski inequality for q-torsional rigidity. As a corollary, we reveal that the functional Orlicz–Brunn–Minkowski inequality is equivalent to
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Rational points close to non-singular algebraic curves Mathematika (IF 0.8) Pub Date : 2023-06-28 Faustin Adiceam, Oscar Marmon
We study the density of solutions to Diophantine inequalities involving non-singular ternary forms, or equivalently, the density of rational points close to non-singular plane algebraic curves.
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On the distribution of equivalence classes of random symmetric p-adic matrices Mathematika (IF 0.8) Pub Date : 2023-06-19 Valeriya Kovaleva
We consider random symmetric matrices with independent entries distributed according to the Haar measure on Z p $\mathbb {Z}_p$ for odd primes p and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that
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On polynomials with only rational roots Mathematika (IF 0.8) Pub Date : 2023-06-09 Lajos Hajdu, Robert Tijdeman, Nóra Varga
In this paper, we study upper bounds for the degrees of polynomials with only rational roots. First, we assume that the coefficients are bounded. In the second theorem, we suppose that the primes 2 and 3 do not divide any coefficient. The third theorem concerns the case that all coefficients are composed of primes from a fixed finite set.
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A pretentious proof of Linnik's estimate for primes in arithmetic progressions Mathematika (IF 0.8) Pub Date : 2023-06-09 Stelios Sachpazis
In the present paper, the author adopts a pretentious approach and recovers an estimate obtained by Linnik for the sums of the von Mangoldt function Λ on arithmetic progressions. It is the analogue of an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the
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Entropic exercises around the Kneser–Poulsen conjecture Mathematika (IF 0.8) Pub Date : 2023-06-06 Gautam Aishwarya, Irfan Alam, Dongbin Li, Sergii Myroshnychenko, Oscar Zatarain-Vera
We develop an information-theoretic approach to study the Kneser–Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map. We answer this question affirmatively in various cases.
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Dimension formulas for Siegel modular forms of level 4 Mathematika (IF 0.8) Pub Date : 2023-05-31 Manami Roy, Ralf Schmidt, Shaoyun Yi
We prove several dimension formulas for spaces of scalar-valued Siegel modular forms of degree 2 with respect to certain congruence subgroups of level 4. In case of cusp forms, all modular forms considered originate from cuspidal automorphic representations of GSp ( 4 , A ) ${\rm GSp}(4,{\mathbb {A}})$ whose local component at p = 2 $p=2$ admits nonzero fixed vectors under the principal congruence
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A note on the zeros of the derivatives of Hardy's function Z(t)$Z(t)$ Mathematika (IF 0.8) Pub Date : 2023-05-30 Hung M. Bui, Richard R. Hall
Using the twisted fourth moment of the Riemann zeta-function, we study large gaps between consecutive zeros of the derivatives of Hardy's function Z ( t ) $Z(t)$ , improving upon previous results of Conrey and Ghosh (J. Lond. Math. Soc. 32 (1985) 193–202), and of the second named author (Acta Arith. 111 (2004) 125–140). We also exhibit small distances between the zeros of Z ( t ) $Z(t)$ and the zeros
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Souplet–Zhang and Hamilton-type gradient estimates for non-linear elliptic equations on smooth metric measure spaces Mathematika (IF 0.8) Pub Date : 2023-05-21 Ali Taheri, Vahideh Vahidifar
In this article, we present new gradient estimates for positive solutions to a class of non-linear elliptic equations involving the f-Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive
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Distribution of Dirichlet L-functions Mathematika (IF 0.8) Pub Date : 2023-05-16 Zikang Dong, Weijia Wang, Hao Zhang
In this article, we study the distribution of values of Dirichlet L-functions, the distribution of values of the random models for Dirichlet L-functions, and the discrepancy between these two kinds of distributions. For each question, we consider the cases of 1 2 < Re s < 1 $\frac{1}{2}<\operatorname{Re}s<1$ and Re s = 1 $\operatorname{Re}s=1$ separately.
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The chromatic number of Rn$\mathbb {R}^{n}$ with multiple forbidden distances Mathematika (IF 0.8) Pub Date : 2023-05-09 Eric Naslund
Let A ⊂ R > 0 $A\subset \mathbb {R}_{>0}$ be a finite set of distances, and let G A ( R n ) $G_{A}(\mathbb {R}^{n})$ be the graph with vertex set R n $\mathbb {R}^{n}$ and edge set { ( x , y ) ∈ R n : ∥ x − y ∥ 2 ∈ A } $\lbrace (x,y)\in \mathbb {R}^{n}:\ \Vert x-y\Vert _{2}\in A\rbrace$ , and let χ ( R n , A ) = χ ( G A ( R n ) ) $\chi (\mathbb {R}^{n},A)=\chi (G_{A}(\mathbb {R}^{n}))$ . Erdős asked
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Curvatures for unions of WDC sets Mathematika (IF 0.8) Pub Date : 2023-05-04 Dušan Pokorný
We prove the existence of the curvature measures for a class of U WDC ${\mathcal {U}}_{{\rm WDC}}$ sets, which is a direct generalisation of U P R ${\mathcal {U}}_{\rm {P\! R}}$ sets studied by Rataj and Zähle. Moreover, we provide a simple characterisation of U WDC ${\mathcal {U}}_{{\rm WDC}}$ sets in R 2 $\mathbb {R}^2$ and prove that in R 2 $\mathbb {R}^2$ , the class of U WDC ${\mathcal {U}}_{{\rm
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Sums of distances on graphs and embeddings into Euclidean space Mathematika (IF 0.8) Pub Date : 2023-04-18 Stefan Steinerberger
Let G = ( V , E ) $G=(V,E)$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices x 1 , ⋯ , x k $x_1, \dots , x_k$ , take x k + 1 $x_{k+1}$ to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges
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Sums of triples in Abelian groups Mathematika (IF 0.8) Pub Date : 2023-04-18 Ido Feldman, Assaf Rinot
Motivated by a problem in additive Ramsey theory, we extend Todorčević's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for every Abelian group G of size ℵ2, there exists a coloring c : G → Z $c:G\rightarrow \mathbb {Z}$ such that for every uncountable X ⊆ G $X\subseteq G$ and every
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The Kℵ0$K^{\aleph _0}$ game: Vertex colouring Mathematika (IF 0.8) Pub Date : 2023-04-14 Nathan Bowler, Marit Emde, Florian Gut
We investigate games played between Maker and Breaker on an infinite complete graph whose vertices are coloured with colours from a given set, each colour appearing infinitely often. The players alternately claim edges, Maker's aim being to claim all edges of a sufficiently colourful infinite complete subgraph and Breaker's aim being to prevent this. We show that if there are only finitely many colours
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On the error term in a mixed moment of L-functions Mathematika (IF 0.8) Pub Date : 2023-04-11 Rizwanur Khan, Zeyuan Zhang
There has recently been some interest in optimizing the error term in the asymptotic for the fourth moment of Dirichlet L-functions and a closely related mixed moment of L-functions involving automorphic L-functions twisted by Dirichlet characters. We obtain an improvement for the error term of the latter.
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On the number of vertices of projective polytopes Mathematika (IF 0.8) Pub Date : 2023-03-23 Natalia García-Colín, Luis Pedro Montejano, Jorge Luis Ramírez Alfonsín
Let X be a set of n points in R d $\mathbb {R}^d$ in general position. What is the maximum number of vertices that conv ( T ( X ) ) $\mathsf {conv}(T(X))$ can have among all the possible permissible projective transformations T? In this paper, we investigate this and other related questions. After presenting several upper bounds, obtained by using oriented matroid machinery, we study a closely related
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Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line Mathematika (IF 0.8) Pub Date : 2023-03-23 Fan Ge
Assuming the Riemann hypothesis and a hypothesis on small gaps between zeta zeros (see equation (ES 2K) below for a precise definition), we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith [J. Math. Phys. 60 (2019), no. 8, 083509], which states that for any positive integer K and real number a>0$a>0$,
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Isoperimetric problems for zonotopes Mathematika (IF 0.8) Pub Date : 2023-03-15 Antal Joós, Zsolt Lángi
Shephard (Canad. J. Math. 26 (1974), 302–321) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note, we prove a generalization of this theorem yielding similar formulae for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric
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On a geometric combination of functions related to Prékopa–Leindler inequality Mathematika (IF 0.8) Pub Date : 2023-03-07 Graziano Crasta, Ilaria Fragalà
We introduce a new operation between nonnegative integrable functions on Rn$\mathbb {R}^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence,
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Unions of lines in Rn$\mathbb {R}^n$ Mathematika (IF 0.8) Pub Date : 2023-02-17 Joshua Zahl
We prove a conjecture of D. Oberlin on the dimension of unions of lines in Rn$\mathbb {R}^n$. If d⩾1$d\geqslant 1$ is an integer, 0⩽β⩽1$0\leqslant \beta \leqslant 1$, and L is a set of lines in Rn$\mathbb {R}^n$ with Hausdorff dimension at least ◂⋅▸2(d−1)+β$2(d-1)+\beta$, then the union of the lines in L has Hausdorff dimension at least d+β$d + \beta$. Our proof combines a refined version of the multilinear
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On Borsuk–Ulam theorems and convex sets Mathematika (IF 0.8) Pub Date : 2023-02-11 M. C. Crabb
The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if ◂,▸f:S1→◂◽˙▸R2k+1$f: S^1\rightarrow {\mathbb {R}}^{2k+1}$ is a continuous function on the unit circle S1 in C${\mathbb {C}}$ such that ◂=▸f(−z)=−f(z)$f(-z)=-f(z)$ for all z∈S1$z\in S^1$, then there is a finite subset X of S1 of diameter at most π−◂+▸π/(2k+1)$\pi -\pi
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The distribution of geodesics on the cube surface Mathematika (IF 0.8) Pub Date : 2023-02-11 Yuxuan Yang
We establish a Kronecker–Weyl type result, on time-quantitative equidistribution for a natural non-integrable system, geodesic flow on the cube surface. Our tool is the shortline-ancestor method developed in Beck, Donders, and Yang [Acta Math. Hungar. 161 (2020), 66–184] and Beck, Donders, and Yang [Acta Math. Hungar. 162 (2020), 220–324], modified in an appropriate way to embrace all slopes. The method
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Extremal values of semi-regular continuants and codings of interval exchange transformations Mathematika (IF 0.8) Pub Date : 2023-02-11 Alessandro De Luca, Marcia Edson, Luca Q. Zamboni
Given a set A$\mathbb {A}$ consisting of positive integers ◂<▸a1
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Affine subspace concentration conditions for centered polytopes Mathematika (IF 0.8) Pub Date : 2023-02-13 Ansgar Freyer, Martin Henk, Christian Kipp
Recently, K.-Y. Wu introduced affine subspace concentration conditions for the cone volumes of polytopes and proved that the cone volumes of centered, reflexive, smooth lattice polytopes satisfy these conditions. We extend the result to arbitrary centered polytopes.
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Average shadowing and gluing property Mathematika (IF 0.8) Pub Date : 2023-02-06 Michael Blank
The purpose of this work is threefold: (i) extend shadowing theory for discontinuous and non-invertible systems, (ii) consider more general classes of perturbations (for example, small only on average), (iii) establish a general theory based on the property that the shadowing holds for the case of a single perturbation. The “gluing” construction used in the analysis of the last property turns out to
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On the number variance of zeta zeros and a conjecture of Berry Mathematika (IF 0.8) Pub Date : 2023-01-25 Meghann Moriah Lugar, Micah B. Milinovich, Emily Quesada-Herrera
Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in
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On strong chains of sets and functions Mathematika (IF 0.8) Pub Date : 2023-01-03 Tanmay C. Inamdar
Shelah has shown that there are no chains of length ω3 increasing modulo finite in ◂⋅▸ω2ω2${}^{\omega _2}\omega _2$. We improve this result to sets. That is, we show that there are no chains of length ω3 in ◂◽˙▸[ω2]ℵ2$[\omega _2]^{\aleph _2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω2 increasing modulo finite in
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Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic L-functions Mathematika (IF 0.8) Pub Date : 2022-12-27 Xiaoguang He, Mengdi Wang
We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that the above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic L-function on ◂⋅▸GLm(m⩾2)$GL_m (m \geqslant 2)$ and polynomial nilsequences has logarithmic
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Exponentially larger affine and projective cap Mathematika (IF 0.8) Pub Date : 2022-12-19 Christian Elsholtz, Gabriel F. Lipnik
In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach and by Ellenberg and Gijswijt), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus p. Moreover, we show that for all primes ◂≡▸p≡5mod6$p \equiv 5 \bmod 6$ with p⩽41$p \leqslant