Acta Arith Research Articles

On the Diophantine equations of the form λ1Un1+λ2Un2+⋯+λkUnk=wp1z1p2z2⋯pszs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _1U_{n_1} + \\lambda _2U_{n_2} +\\cdots + \\lambda _kU_{n_k} = wp_1^{z_1}p_2^{z_2} \\cdots p_s^{z_s}$$\\end{document}

In this paper, we consider the Diophantine equation λ1Un1+⋯+λkUnk=wp1z1…pszs,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _1U_{n_1}+\\cdots +\\lambda _kU_{n_k}=wp_1^{z_1} \\ldots p_s^{z_s},$$\\end{document} where {Un}n≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{U_n\\}_{n\\ge 0}$$\\end{document} is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; w is a fixed non-zero integer; p1,⋯,ps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_1,\\dots ,p_s$$\\end{document} are fixed, distinct prime numbers; λ1,⋯,λk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _1,\\dots ,\\lambda _k$$\\end{document} are strictly positive integers; and n1,⋯,nk,z1,⋯,zs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1,\\dots ,n_k,z_1,\\dots ,z_s$$\\end{document} are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions (n1,⋯,nk,z1,⋯,zs)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n_1,\\dots ,n_k,z_1,\\dots ,z_s)$$\\end{document}. In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (Monatshefte Math 185(1):103–131, 2018), Mazumdar and Rout (Monatshefte Math 189(4):695–714, 2019), Meher and Rout (Lith Math J 57(4):506–520, 2017), and Ziegler (Acta Arith 190:139–169, 2019).

ON THE LOWEST ZERO OF THE DEDEKIND ZETA FUNCTION

Abstract Let $\zeta _K(s)$ denote the Dedekind zeta-function associated to a number field K. We give an effective upper bound for the height of the first nontrivial zero other than $1/2$ of $\zeta _K(s)$ under the generalised Riemann hypothesis. This is a refinement of the earlier bound obtained by Sami [‘Majoration du premier zéro de la fonction zêta de Dedekind’, Acta Arith.99(1) (2000), 61–65].

ℓ-adic digits and class number of imaginary quadratic fields

Motivated by the work of [K. Girstmair, A “popular” class number formula, Amer. Math. Monthly 101(10) (1994) 997–1001; K. Girstmair, The digits of [Formula: see text] in connection with class number factors, Acta Arith. 67(4) (1994) 381–386] and [M. R. Murty and R. Thangadurai, The class number of [Formula: see text] and digits of [Formula: see text], Proc. Amer. Math. Soc. 139(4) (2010) 1277–1289], we study the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] whenever [Formula: see text] is a product of two distinct primes or a prime power. More explicitly, if [Formula: see text] is an integer such that [Formula: see text] and suppose that [Formula: see text] is the [Formula: see text]-adic expansion of [Formula: see text] then we establish the average of the digits of the [Formula: see text]-adic expansion of [Formula: see text] in terms of [Formula: see text] and the “trace” of generalized Bernoulli numbers [Formula: see text] where [Formula: see text]’s are odd Dirichlet characters modulo [Formula: see text] As a consequence of these results, we recover two well-known results of Gauss and Heilbronn (see Theorems 1.6 and 1.7).

ARITHMETIC PROPERTIES OF AN ANALOGUE OF t-CORE PARTITIONS

Abstract An integer partition of a positive integer n is called t-core if none of its hook lengths is divisible by t. Gireesh et al. [‘A new analogue of t-core partitions’, Acta Arith.199 (2021), 33–53] introduced an analogue $\overline {a}_t(n)$ of the t-core partition function. They obtained multiplicative formulae and arithmetic identities for $\overline {a}_t(n)$ where $t \in \{3,4,5,8\}$ and studied the arithmetic density of $\overline {a}_t(n)$ modulo $p_i^{\,j}$ where $t=p_1^{a_1}\cdots p_m^{a_m}$ and $p_i\geq 5$ are primes. Bandyopadhyay and Baruah [‘Arithmetic identities for some analogs of the 5-core partition function’, J. Integer Seq.27 (2024), Article no. 24.4.5] proved new arithmetic identities satisfied by $\overline {a}_5(n)$ . We study the arithmetic densities of $\overline {a}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^\alpha m$ where $\gcd (m,6)$ =1. Also, employing a result of Ono and Taguchi [‘2-adic properties of certain modular forms and their applications to arithmetic functions’, Int. J. Number Theory1 (2005), 75–101] on the nilpotency of Hecke operators, we prove an infinite family of congruences for $\overline {a}_3(n)$ modulo arbitrary powers of 2.

The Dimension of the Set of ψ-Badly Approximable Points in All Ambient Dimensions: On a Question of Beresnevich and Velani

Abstract Let $\psi :{\mathbb{N}} \to [0,\infty )$, $\psi (q)=q^{-(1+\tau )}$ and let $\psi $-badly approximable points be those vectors in ${\mathbb{R}}^{d}$ that are $\psi $-well approximable, but not $c\psi $-well approximable for arbitrarily small constants $c>0$. We establish that the $\psi $-badly approximable points have the Hausdorff dimension of the $\psi $-well approximable points, the dimension taking the value $(d+1)/(\tau +1)$ familiar from theorems of Besicovitch and Jarník. The method of proof is an entirely new take on the Mass Transference Principle (MTP) by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named “delayed pruning” to construct a sufficiently large $\liminf $ set and combine this with ideas inspired by the proof of the MTP to find a large $\limsup $ subset of the $\liminf $ set. Our results are a generalisation of some $1$-dimensional results due to Bugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothing alike.

On the abc$abc$ conjecture in algebraic number fields

AbstractIn this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , 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 bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

A dichotomy phenomenon for bad minus normed Dirichlet

AbstractGiven a norm ν on , the set of ν‐Dirichlet improvable numbers 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, , where is the set of badly approximable numbers. Each of the sets , like , is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either or else has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether the critical locus of ν intersects a precompact ‐orbit, where is the one‐parameter diagonal subgroup of acting on the space X of unimodular lattices in . Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice , either is unbounded (and then any precompact ‐orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices in X whose ‐trajectories are precompact and contain Λ in their closure has full Hausdorff dimension.

A note on the zeros of the derivatives of Hardy's function Z(t)$Z(t)$

AbstractUsing the twisted fourth moment of the Riemann zeta‐function, we study large gaps between consecutive zeros of the derivatives of Hardy's function , 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 and the zeros of for every , in support of our numerical observation that the zeros of and , when k and ℓ have the same parity, seem to come in pairs that are very close to each other. The latter result is obtained using the mollified discrete second moment of the Riemann zeta‐function.

Corrigendum to “Explicit estimates for the summatory function of $\varLambda (n)/n$ from the one of $\varLambda(n)$” (Acta Arith. 159 (2013), 113–122)

Corrigendum to “Explicit estimates for the summatory function of $\varLambda (n)/n$ from the one of $\varLambda(n)$” (Acta Arith. 159 (2013), 113–122)

The large sieve with power moduli in imaginary quadratic number fields

In this paper, we establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal [S. Baier and A. Bansal, The large sieve with power moduli for [Formula: see text], Int. J. Number Theory 14 (10) (2018) 2737–2756; Large sieve with sparse sets of moduli for [Formula: see text], Acta Arith. 196 (1) (2020) 17–34] for the Gaussian field.

Realization of modular Galois representations in the Jacobians of modular curves

In Tian (Acta Arith. 164:399–412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod \(\ell \) Galois representations associated to eigenforms f for the cases that \(\ell \ge k-1\) and f has level one, where k is the weight of f. In this paper, we generalize the results of Tian (Acta Arith. 164:399–412, 2014) and present a method to find the Jacobians of modular curves of minimal dimensions to realize the modular Galois representations. Our method works for the cases that \(\ell \ge 5\) may be any prime without the assumption \(\ell \ge k-1\) and the eigenforms f have arbitrary levels prime to \(\ell \). Moreover, if \(k>2\), we give criteria for realizing the mod \(\ell \) Galois representations in the Jacobians \(J_0\) of \(X_0\).

Decoupling for fractal subsets of the parabola

We consider decoupling for a fractal subset of the parabola. We reduce studying \(l^{2}L^{p}\) decoupling for a fractal subset on the parabola \(\{(t, t^2) : 0 \le t \le 1\}\) to studying \(l^{2}L^{p/3}\) decoupling for the projection of this subset to the interval [0, 1]. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain–Demeter’s decoupling theorem for the parabola. In the case when p/3 is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to [0, 1]. Our ideas are inspired by the recent work on ellipsephic sets by Biggs (arXiv:1912.04351, 2019 and Acta Arith. 200(4):331–348, 2021) using nested efficient congruencing.

Errata to “On second 2-descent and non-congruent numbers” (Acta Arith. 170 (2015), 343–360)

Errata to “On second 2-descent and non-congruent numbers” (Acta Arith. 170 (2015), 343–360)

Families of congruences for fractional partition functions modulo powers of primes

Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.

Discrepancy bounds for a class of negatively dependent random points including Latin hypercube samples

We introduce a class of γ-negatively dependent random samples. We prove that this class includes, apart from Monte Carlo samples, in particular Latin hypercube samples and Latin hypercube samples padded by Monte Carlo. For a γ-negatively dependent N-point sample in dimension d we provide probabilistic upper bounds for its star discrepancy with explicitly stated dependence on N, d, and γ. These bounds generalize the probabilistic bounds for Monte Carlo samples from Heinrich et al. (Acta Arith. 96 (2001) 279–302) and C. Aistleitner (J. Complexity 27 (2011) 531–540), and they are optimal for Monte Carlo and Latin hypercube samples. In the special case of Monte Carlo samples the constants that appear in our bounds improve substantially on the constants presented in the latter paper and in C. Aistleitner and M. T. Hofer (Math. Comp. 83 (2014) 1373–1381).

A representative of RΓ(N,T) for higher dimensional twists of ℤpr(1)

Let [Formula: see text] be a Galois extension of [Formula: see text]-adic number fields and let [Formula: see text] be a de Rham representation of the absolute Galois group [Formula: see text] of [Formula: see text]. In the case [Formula: see text], the equivariant local [Formula: see text]-constant conjecture describes the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin [Formula: see text]-functions and it can be formulated as the vanishing of a certain element [Formula: see text] in [Formula: see text]; a similar approach can be followed also in the case of unramified twists [Formula: see text] of [Formula: see text] (see [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383; D. Izychev and O. Venjakob, Equivariant epsilon conjecture for 1-dimensional Lubin–Tate groups, J. Théor. Nr. Bordx. 28(2) (2016) 485–521]). One of the main technical difficulties in the computation of [Formula: see text] arises from the so-called cohomological term [Formula: see text], which requires the construction of a bounded complex [Formula: see text] of cohomologically trivial modules which represents [Formula: see text] for a full [Formula: see text]-stable [Formula: see text]-sublattice [Formula: see text] of [Formula: see text]. In this paper, we generalize the construction of [Formula: see text] in Theorem 2 of [W. Bley and A. Cobbe, The equivariant local [Formula: see text]-constant conjecture for unramified twists of [Formula: see text], Acta Arith. 178(4) (2017) 313–383] to the case of a higher dimensional [Formula: see text].

The 2-Sylow subgroup of $$K_2O_F$$ for certain quadratic number fields

In this paper, we apply Qin’s theorem for the 4-rank of $$K_2O_F$$ to establish the relation between the 4-rank of the ideal class group of $$F=\mathbb {Q}(\sqrt{d})$$ and the 4-rank of $$K_2O_F$$ provided that all odd prime factors of d are congruent to 1 mod 8. As an application, we give a concise and unified proof of two conjectures proposed by Conner and Hurrelbrink (Acta Arith 73:59–65, 1995).

An elementary proof for the number of supersingular elliptic curves

Building on Finotti in (Acta Arith 139(3):265–273, 2009), we give an elementary proof for the well known result that there exactly $$\lceil (p-1)/4 \rceil -\lfloor (p-1)/6 \rfloor$$ supersingular elliptic curves in characteristic p. We use a related polynomial instead of the supersingular polynomial itself to simplify the proof and this idea might be helpful to prove other results related to the supersingular polynomial.

Discrete Correlation of Order 2 of Generalized Rudin-Shapiro Sequences on Alphabets of Arbitrary Size

Abstract In 2009, Grant, Shallit, and Stoll [Acta Arith. 140 (2009), [345–368] constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al.

Relative bifurcation sets and the local dimension of univoque bases

AbstractFix an alphabet$A=\{0,1,\ldots ,M\}$with$M\in \mathbb{N}$. The univoque set$\mathscr{U}$of bases$q\in (1,M+1)$in which the number$1$has a unique expansion over the alphabet$A$has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set$\mathscr{U}$are distributed over the interval$(1,M+1)$by determining the limit$$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$for all$q\in (1,M+1)$. We show in particular that$f(q)>0$if and only if$q\in \overline{\mathscr{U}}\backslash \mathscr{C}$, where$\mathscr{C}$is an uncountable set of Hausdorff dimension zero, and$f$is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of$\mathscr{U}$called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of$\mathscr{U}$with any interval, answering a question of Kalleet al [On the bifurcation set of unique expansions. Acta Arith.188(2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets.Adv. Math.308(2017), 575–598] on strongly univoque sets.