Papers

The Tchoukaillon array is an infinite array of the positive integers, arising from a one-row Mancala solitaire, in which each positive integer occurs exactly once. Its zeroth column is the Flavius Josephus sieve and its zeroth row is the sequence of Tchoukaillon numbers; the asymptotics of these two edges are classical results of Andersson and of Broline and Loeb. On the basis of numerical evidence, N. Beluhov conjectured (as relayed by Knuth) that the general entry $T_{i,j}$ satisfies $T_{i,j} \approx (\pi i+2j)^2/(4\pi)$ as $i,j \to \infty$. We prove this conjecture. In fact we establish the stronger uniform estimate $T_{i,j} = (\pi i+2j+2)^2/(4\pi) + O((i+j+1)^{4/3})$, in which both constants $\pi$ and $2$ are produced by the array's own recursion through a Wallis product, independently of the two edge theorems. Equivalently, the square root of the entry is asymptotically linear, $\sqrt{T_{i,j}} = (\sqrt{\pi}/2)\, i + (1/\sqrt{\pi})(j+1) + O((i+j+1)^{1/3})$, the linear blend of the two edge growth-rates. As corollaries we obtain that the level regions $\{T_{i,j} \le V\}$ are triangles up to a boundary of width $O(V^{1/6})$, and an $O(\sqrt{M})$ algorithm that locates the row and column of a given integer $M$.

In this paper, we investigate the underlying algebraic and combinatorial structures governing the coset representatives (shift vectors) for Construction A of lattices, with a particular focus on equations (14) and (15) presented in Paragraph 3, Chapter 8 of the seminal work by J.H. Conway and N.J.A. Sloane, "Sphere Packings, Lattices and Groups". These dual equations define the boundary conditions for the analytical generation of the 64-dimensional Quebbemann lattice (Q64). We prove that seeking non-zero solutions constrained by arithmetic or geometric progressions yields a structural collapse to the trivial zero vector due to the transcendental nature of {\pi}. Conversely, by relaxing these bounds to unique coordinate configurations, we uncover an exact, closed-form algebraic core governed by the alternating coefficients of the Pascal triangle. Furthermore, we implement an energy-minimization model via continuous-to-discrete projection that yields an optimal shift vector with a strictly integer Euclidean norm ||z||^2 = 20.000000. Finally, we bridge this formulation to the Repeated Differences paradigm of Craig's lattices Anm, showing how the spectral components of the {\Theta}-series are naturally filtered by these binomial structures.

In an influential paper [K. Ribet, Bimodules and abelian surfaces, in Algebraic number theory, 359-407, Adv. Stud. Pure Math., Vol. 17, 1989] on the bad reduction of Shimura curves, Ribet studies certain superspecial abelian surfaces over $\overline{\mathbb{F}}_p$ with quaternion multiplication by a maximal order $\mathcal{O}$ in an indefinite quaternion $\mathbb{Q}$-algebra ramified at $p$. In particular, he classifies the $p$-divisible groups of such $\mathcal{O}$-abelian surfaces by classifying $(\mathcal{O}_p, \mathcal{O}_p)$-bimodules $L_p$ that are free over $\mathbb{Z}_p$ (i.e.bilattices) under an additional admissible assumption. In this paper, we generalize Ribet's result by removing the admissible assumption and producing a complete classification of $(\mathcal{O}_p, \mathcal{O}_p)$-bilattices $L_p$. Equip the right order $\mathcal{O}_p$ with the canonical involution, and suppose additionally that the left order $\mathcal{O}_p$ is equipped with an orthogonal involution $*$. We derive the necessary and sufficient condition for the existence of a perfect quaternion hermitian form $\langle~,~\rangle_p:L_p\times L_p\to \mathcal{O}_p$ on the right $\mathcal{O}_p$-lattice $L_p$ inducing the given involution $*$ on the left order $\mathcal{O}_p$, and give a complete classification of such self-dual quaternion hermitian $(\mathcal{O}_p, *, \mathcal{O}_p)$-bilattices $(L_p, \langle~,~\rangle_p)$. Globally, we apply these classification results to the study of the existence of principal polarizations on superspecial abelian varieties over $\overline{\mathbb{F}}_p$ equipped with $\mathcal{O}$-action.

We show that, for Weil–Petersson random closed hyperbolic surfaces of large genus, the normalized weighted count of prime geodesics with norm in the interval $(X,X+H]$ is asymptotically Gaussian, provided $H \leq X$ and $H/\log X\to\infty$ as $X\to\infty$. In particular, it applies to intervals which are not necessarily short.

In a remarkable breakthrough, Faulhuber, Petersen, van Velthoven, and Voigtlaender disproved the Heil–Ramanathan–Topiwala conjecture by constructing twelve linearly dependent time–frequency shifts of a Schwartz function. The following day, the author compressed their vector-Zak and cohomological architecture to an explicit intrinsically subcritical four-point configuration, thereby attaining the minimum possible cardinality. Soon thereafter, Jasper and Mixon gave a transparent analytic refinement of the breakthrough mechanism and proved, for every Diophantine pair, that HRT counterexamples may be realized with windows in the Roumieu Gelfand–Shilov class $\mathcal S_1^1(\mathbb R)$; their theorem establishes the endpoint regularity phenomenon in broad arithmetic generality while leaving support cardinality unrestricted. The purpose of the present paper is to show that these two sharp advances can be attained simultaneously. We prove that the same explicit four points, coefficients, and matrix cocycle admit a nonzero window in $\mathcal S_1^1(\mathbb R)$. Moreover, if $0\ne f\in\mathcal S_s^\sigma(\mathbb R)$ and $s<1$ or $\sigma<1$, then every finite system of distinct time–frequency shifts of $f$ is linearly independent. Thus four is the least possible cardinality and $(1,1)$ is the sharp coordinatewise threshold within the Roumieu Gelfand–Shilov scale.

In this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real closed fields and quadratic forms. We show, by means of Galois theory, that $F(\sqrt{-1})$ is algebraically closed if $F$ is real-closed. Lastly, we explain the algebraic closure of $\mathbb R(\sqrt{-1})=\mathbb C$ by demonstrating the real-closeness of $\mathbb R$.

For a formally real field $F$, we study totally positive field extensions $K$ over $F$. We show that, if $K /F$ is Galois and totally positive then so is the corresponding extension of their pythagorean closures $K_{\rm py}$ over $F_{\rm py}$. We also study the behaviour of weak isotropy and weak hyperbolicity of central simple algebras with an orthogonal involution over totally positive field extensions. We use some of these results to prove new cases for which a conjecture due to Becher holds.

We establish a uniform asymptotic formula for the smoothly weighted shifted second moment of the Gaussian angular Hecke $L$-functions $L(s,\lambda^k)$. The completed moment is expressed as the sum of four explicit main terms, corresponding to the four functional-equation swaps, with an error of size $O_{\Phi,\epsilon}(K^{1/2+\epsilon})$.After the Archimedean factors are removed, the resulting formula agrees with the numerator-only specialization of the four-swap prediction of the $L$-functions Ratios Conjecture. The proof transforms the off-diagonal contribution into Weyl sums over the roots of $r^2\equiv-1\pmod C$, realizes these sums spectrally through incomplete Poincaré series evaluated at $i$, and separates the Eisenstein and Maaß spectra. The two $v$-type main terms arise respectively from the zero frequency and from the combined residues of two moving Eisenstein poles, while the cuspidal spectrum is absorbed into the square-root error term.

Over any infinite field we prove existence of polynomial automorphism with prescribed differentials at points. More precisely, let $\K$ be an infinite field and $a_1,\ldots, a_k;$ $b_1,\ldots , b_k$ be two sequences of different points in~$\K^n$, $n\ge 2$. For any sequence $L_1,\ldots, L_k\in SL(n,\K)$ there exists a polynomial automorphism $\Phi: \K^n\to \K^n$ with jacobian one such that $\Phi(a_i)=b_i$ and $d_{a_i} \Phi= L_i$ for $i=1,\ldots, k.$

Let $K$ be an imaginary quadratic number field, let $p$ be an odd prime, and let $G=G_{\varnothing}(K)(p)$ be the Galois group of the maximal everywhere unramified pro-$p$ extension of $K$. To each mod-$p$ character $x$ of $G$ we associate a linear map $D_x$ from $\mathrm{Cl}(K)[p]$ to $\mathrm{Cl}(K)/p$; a formula of Ahlqvist and Carlson expresses it through the class of a norm ideal in the unramified cyclic degree-$p$ extension attached to $x$. These maps determine all triple Massey products on $H^1(G,\mathbb F_p)$, and with them the cubic initial relations of $G$. Suppose that the $p$-class rank $d=\dim_{\mathbb F_p}\mathrm{Cl}(K)/p$ is at least three. The $(d-1)\times(d-1)$ minors of the family $x\mapsto D_x$ define a subscheme $\Sigma_D$ of $\mathbb P^{d-1}_{\mathbb F_p}$, an invariant of $K$, the norm-degeneracy scheme. We prove: if the rank condition $\mathrm{rk}\,D_x=d-2$ holds transversally at a point of $\Sigma_D$, over some finite extension of $\mathbb F_p$, then $G$ is mild, and hence of cohomological dimension 2. For $p>3$ transversality means that $\Sigma_D$ is smooth of dimension $d-3$ at the point; at $p=3$ the kernel of the Bockstein map enters as an additional constraint. We treat every imaginary quadratic field of $p$-class rank at least three with $|D_K|<2^{30}$. Only $p=3$, 5, and 7 occur. The criterion decides 206 of the 207 fields at $p=5$ and 7, and 505 of the 12 750 fields at $p=3$, where the Bockstein condition restricts its reach. A direct computation with the cubic initial relations settles the remaining field at $p=5$ and a further 11 765 at $p=3$, 26 of them not mild, while 480 remain undecided. In all, the $p$-class tower group is proved mild for 12 451 of the 12 957 fields. These appear to be the first number fields for which the full maximal everywhere unramified pro-$p$ Galois group is proved to be mild, and hence of cohomological dimension 2.

The classical sphere packing problem, which remains unsolved, consists of determining how densely a large number of identical spheres can be packed together. In some sphere packings, the centers of the spheres in a sphere packing form a Euclidean lattice, which is a discrete additive subgroup of $\mathbb{R}^n$. Free $\mathbb{Z}$-modules in the ring of integers of an algebraic number field yield algebraic lattices via the canonical embedding. In this work, we present new constructions of algebraic lattices from certain families of $\mathbb{Z}$-modules in the ring of algebraic integers of subfields of the $p$-th cyclotomic field, where $p$ is a prime number. Within this framework, we compute lower bounds for the center density of these algebraic lattices and construct algebraic lattices having the best known packing density in dimensions 2, 3, and 5.

Let $d(n)$ be the divisor function and let $H<H'$ be consecutive highly composite numbers. We study directed unit moves between their prime-exponent vectors under the hard ceiling $z\le H'$. The normalised capacity of such a geodesic is its smallest divisor count divided by $d(H)$, giving a finite fixed-endpoint maximin problem. The exact record enumeration first finds a failure of the static surrogate $d(\gcd(H,H'))\ge d(H)/2$ at $48,886,437,600<64,250,746,560$, where the ratio is $4/9$. For every state $z$ in the exponent box, however, we prove $$d(z)d(HH'/z)\ge d(H)d(H')$$ and deduce the record-box gap: no box state lies numerically strictly between the two records. We also give an exact dynamic-programming recursion, solve the strata $L_-\le 1$, and reduce the complete $L_-=2$ problem to an explicit divisor-selection functional. A computer-assisted enumeration through $10^{70}$ produces $889$ records and $888$ transitions. Although the static half-gcd bound fails $119$ times, every computed geodesic capacity is at least $1/2$; equality occurs in exactly the $124$ transitions that lose an exponent-one support prime. Independently checked certificates cover all $301$ transitions with $L_-=2$. The corresponding universal half-capacity bound and equality classification remain open beyond the proved strata and the verified range. COMMENTS

The $\Lambda$-distribution of a compact matrix group is an invariant in algebraic probability theory that was recently introduced to study zero distributions of function field $L$-functions. It is encoded by the $\sigma$-moment generating function, a generalization of the Molien series of classical invariant theory. In this work, we compute the $\sigma$-moment generating functions of finite matrix groups in many new cases. In particular, we compute the asymptotic $\Lambda$-distributions for the infinite families of Weyl reflection groups of types $B_n/C_n$ and $D_n$, complementing the previously known case of reflection groups of type $A_n$, i.e., symmetric groups. We also establish a general result relating the shapes of $\sigma$-moment generating functions to the distributions of associated classical random variables, explaining a previous ad hoc observation for independent Gaussians arising from traces of powers on compact classical groups.

We prove some conjectures of K. C. Au concerning the descents of cyclotomic levels in certain sums over multiple polylogarithms. Via answers to a question of Au in some particular cases, we also confirm Broadhurst's conjecture on honorary multiple zeta values at even weights.

Let $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for the $L^1$-mean of the Möbius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an $\ell^2$-estimate involving the Möbius function.

Let $\mathcal{L}_n \subset \mathcal{M}_2$ be the locus of genus-two curves admitting a maximal degree-$n$ elliptic subcover, cut out in $\mathbb{P}(2,4,6,10)$ by an irreducible weighted-homogeneous polynomial $F_n \in \mathbb{Z}[J_2,J_4,J_6,J_{10}]$. Let $\nu(n)$ be the degree of $X_1(n) \to X(1)$, let $G_{n^2}$ be the Siegel modular form of level one with divisor the Humbert surface $H_{n^2}$, and let $k(H_{n^2})$ be its weight. We prove that the meromorphic Siegel modular form $F_n(\tau)$ obtained from $F_n$ has a pole of order exactly $\nu(n)$ along the product locus, that $\chi_{10}^{\nu(n)} F_n(\tau)$ is a constant multiple of $G_{n^2}$, and that $\deg_w F_n = k(H_{n^2}) - 10\nu(n)$ for every $n \geq 2$, even or odd. We determine the restriction of $G_{n^2}$ to the product locus as an explicit product of modular polynomials and, for $n \geq 3$, its leading Fourier-Jacobi coefficient as a product of theta functions over the torsion points of exact order $n$, and we characterize $G_{n^2}$, up to scalar, as the unique form of its weight vanishing on a single torsion divisor. These data convert the computation of $F_n$ from elimination into a linear problem of the size the formula prescribes, which we carry out for $n=5$.

Oort's conjecture asserts that for any $g \geq 2$ and any prime $p$, every geometric generic member in the supersingular locus $\mathcal{S}_g$ has automorphism group $\{ \pm 1 \}$. This has been proved very recently by Viehmann in full generality, with previously known counterexamples for $p=2$ and $g=2,3$. We construct, for any $g \geq 3$, a closed subvariety of dimension $g-1$ which contains an open dense subset $\mathcal{U}$ of $a$-invariant $g-2$ such that $\mathcal{U}$ meets every irreducible component of $\mathcal{S}_g$ and every geometric point in $\mathcal{U}$ has automorphism group $\{ \pm 1 \}$ if $p >2$. This gives an independent proof of Oort's conjecture for $p>2$. When $g>3$, so $a = g-2 > 1$, this provides complementary information to the $a=1$ locus investigated in Viehmann's work on how the automorphism groups interact with the geometry.

For $N\geq 2$ and $k\geq 1$, let $M_k(N):=\#\{x_1\cdots x_k : x_i\in\{1,\ldots,N\}\text{ for all } i\}$ be the $k$-dimensional multiplication table. Given $N$, Khovanskii's theorem implies that $M_k(N)$ agrees, for all sufficiently large $k$, with a polynomial in $k$ of degree $\pi(N)$. We determine the asymptotic size of its leading coefficient, proving that, as $N\to\infty$, with $k$ sufficiently large relative to $N$, \[ M_k(N) = \exp\bigg((2\pi+o(1))\frac{\sqrt{N}}{\log N}\bigg)\frac{k^{\pi(N)}}{\pi(N)!}. \] We also study the analogous problem when the factors are restricted to $y$-smooth integers. For $y=o(\log N)$, we prove that the number of distinct products of $k$ such integers up to $N$ is asymptotic to the number of $y$-smooth integers up to $N^k$, uniformly for $k\geq 1$.

Let $k\ge1$ be an integer and let $\lambda$ be the Liouville function. In 1965, Chowla gave a conjecture that the values of $\lambda(n+h_1),\dots, \lambda(n+h_k)$ are asymptotically unrelated for any distinct natural numbers $h_1, \dots, h_k$. In this article, motivated by the recent work of Bergelson and Richter on the dynamical generalizations of the prime number theorem, we will show a dynamical generalization of Chowla's conjecture on average. In the proof, we follow an approach of Qi and Zheng who established a variant of Bergelson and Richter's theorem over irreducible binary cubic forms. Moreover, we will use this approach to show an analogue of the dynamical Chowla's conjecture along the primes on average. In 2016, Tao proved that the two-point logarithmic Chowla's conjecture holds. Recently, Charamaras and Richter generalized Tao's theorem to bounded arithmetic functions and proposed a conjecture that generalizes Chowla's conjecture to bounded multi-variable arithmetic functions. Building on their work, we prove a dynamical generalization of Tao's theorem and a variant for the composition of the sum-of-digits function with the prime Omega function.

Let $X$ be the Shimura curve attached to an indefinite quaternion $\mathbb{Q}$-algebra $B$ with a maximal order $O_B$. This paper investigates the reduction $X\otimes \mathbb{F}_p$ of $X$ modulo an arbitrary prime $p$, focusing particularly on its superspecial locus. We give an explicit criterion for the existence of superspecial $\mathbb{F}_q$-rational points on $X$. Furthermore, we compute both the number of geometric superspecial points and the number of $\mathbb{F}_p$-rational superspecial points, through the Eichler class number formula and the Selberg trace formula. As a key ingredient, we classify the Dieudonné modules attached to superspecial $O_B$-abelian surfaces, which generalizes Ribet's classification of admissible quaternion bimodules of rank $2$ by dropping the admissible hypothesis. These results generalize Deuring's explicit formula for supersingular elliptic curves over $\mathbb{F}_p$ and give the Shimura-curve analogue of the Ibukiyama-Katsura formulas for principally polarized superspecial abelian surfaces over $\mathbb{F}_p$.

Fix \(\eta>0\) and \(A_0>0\). Let \(q\) tend to infinity through odd primes, put \(Q=\log q\), and let \(T=T(q)\) satisfy \[ \frac{(\log Q)^{1+\eta}}{Q}\le T\le Q^{A_0}. \] Set \(I=(T,2T]\), and sum without weights over the \(q-2\) nonprincipal characters modulo \(q\). Let \(\mathcal N_q\) count nontrivial zeros in \(I\) with multiplicity, let \(\mathcal N^s_{0,q}\) and \(\mathcal N^*_{0,q}\) count simple and distinct zeros on the critical line, and let \(\mathcal N_{d,q}\) count all distinct zeros in \(I\). Uniformly in this height range, we prove unconditionally \[ \mathcal N_q=\frac{qT\log q}{2\pi}\{1+o_{\eta,A_0}(1)\}, \quad \frac{\mathcal N^s_{0,q}}{\mathcal N_q}, \frac{\mathcal N^*_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}}-o_{\eta,A_0}(1), \quad \frac{\mathcal N_{d,q}}{\mathcal N_q}\ge C_d-o_{\eta,A_0}(1), \] where \[ C_{\mathrm{MT}}=\frac32-\frac1{\sqrt2}\cot\!\left(\frac1{\sqrt2}\right) =0.672500703679\ldots, \qquad C_d=\frac{1+C_{\mathrm{MT}}}{2}=0.836250351839\ldots. \] The proof combines Selberg's family-averaged argument estimate and zero-density deletion with a finite Gevrey Gabor compression of Weil's Hermitian form. Quantitative Fourier–Laplace estimates control exterior zeros down to the lower endpoint, while a local–remote shell decomposition gives uniform control throughout the polylogarithmic upper range. Matrix moment estimates and an inertia-based rank–trace inequality then yield the counting bounds. No form of the generalized Riemann hypothesis is assumed.

We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $\Delta_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $\Delta_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $\rho$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice $\mathbb{Z}^n$, under some natural and necessary congruence restrictions on the scales $\lambda$ at which the set $A$ can contain an isometric copy of the simplex.

We investigate the residual Eisenstein cohomology of an arbitrary unitary group $U(V)$ attached to an arbitrary quadratic extension of number fields $E/F$. Our focus lies on the contribution of the maximal parabolic $F$-subgroups of $U(V)$, for which we identify the cohomologically relevant poles of Eisenstein series and prove that the resulting residues all survive as non-trivial classes in automorphic cohomology in an explicit degree. To illustrate the range of phenomena involved, we study in detail the case of a unitary group over $F=\mathbb{Q}(\sqrt[3]{2})$ of $F$-rank $3$, for which we explicitly construct cuspidal automorphic representations, which satisfy all the assumptions of our main theorem and hence explicitly construct non-zero residual Eisenstein cohomology classes for this unitary group. The methods used in this construction are paradigmatic however, i.e., generalize to other unitary groups over other ground fields $F$ by the use of base change.

Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a finite effectively computable set. This result partially resolves an open problem posed by Tinková, Yatsyna, and the first author. We illustrate our method by explicitly computing $N_K$ for the smallest totally real quintic field $K$ with Galois group $S_5$.

We prove that algebraic values of bounded degree and polynomially bounded height are sparse on rational translates of the graph of a transcendental entire function. More precisely, for fixed $\theta\in\overline{\mathbb{Q}}\cap\mathbb{R}$, $D\geq1$, $t>0$, and for every $\varepsilon>0$, only $O(Q^{\varepsilon})$ rationals $r\in[0,1]$ of height at most $Q$ can satisfy simultaneously $[\mathbb{Q}(f(\theta+r)):\mathbb{Q}]\leq D$ and $H(f(\theta+r))\ll H(r)^t$. Consequently, if these bounds hold for every rational $r$ of sufficiently large height, then $f\in\overline{\mathbb{Q}}[z]$ and $\deg f\leq t$. As applications, we obtain rigidity results for entire functions taking rational or bounded-degree algebraic values with polynomially controlled arithmetic height. In particular, this excludes the polynomial-denominator scenario that arises naturally in connection with Mahler's problem on Liouville numbers. The proof combines Pila's bounded-degree counting theorem with standard height estimates and a simple geometric analysis of transcendental entire graphs.

We study the Hausdorff dimension of the difference sets between the set of typical numbers, defined via the Erdős–Rényi law governing the longest run of digits, and the set of numbers normal in base $2$. Although both properties hold for Lebesgue-almost every real number, neither implies the other. The resulting difference sets $\textit{TpcN}\setminus\textit{Normal}$ and $\textit{Normal}\setminus\textit{TpcN}$ are $D_2({\bf \Pi}_3^0)$-complete in the Borel hierarchy. We show that this logical complexity is matched geometrically: both difference sets have full Hausdorff dimension. Unlike the classical Besicovitch-type sets governing single-digit frequencies, normality requires the correct frequency of every finite block simultaneously, and the explicit Moran-set constructions that succeed for simple normality break down under this requirement. We instead establish the dimension via a counting argument bypassing the need for an explicit construction.

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

We prove a lower bound for the Hausdorff dimension of weighted totally irrational singular vectors in affine-spanning common-base integral self-similar sets satisfying the open set condition. For the middle-third Cantor square, the bound improves the previously known explicit lower bound.

In this paper, we investigate whether Harper's better than square-root cancellation phenomenon still appears in Piatetski-Shapiro sequences, and when one should expect only square-root size. If $c=c(X)$ satisfies $(c-1)\log X\to 0$ as $X\to\infty$, then for Steinhaus or Rademacher random multiplicative functions we have \[ \mathbb{E}\left|\sum_{n\in\mathcal{N}^{(c)}(X)}f(n)\right|=o\!\left(\sqrt{|\mathcal{N}^{(c)}(X)|}\right). \] For Lebesgue almost every fixed $1<c<\frac{8}{7}$, we prove the multiplicative energy satisfies \[ E^\times(\mathcal{N}^{(c)}(X))=2|\mathcal{N}^{(c)}(X)|^2-|\mathcal{N}^{(c)}(X)|+o\left(|\mathcal{N}^{(c)}(X)|^2\right). \] Together with a largest prime factor estimate, this allows us to apply the criterion of Soundararajan and Xu and obtain a standard complex Gaussian limit for the normalized Steinhaus sum. To the best of our knowledge, this gives the first asymptotic formula for the multiplicative energy of Piatetski-Shapiro sequences. For fixed exponent argument, our proof combines metric arguments with the Van der Corput's $B$-process and a double large sieve estimate with Robert-Sargos spacing theorem.

In this paper, we investigate large values of quadratic Dirichlet character sums. We prove new Omega results for both short and long quadratic character sums under the assumption of the Generalized Riemann Hypothesis (GRH), which improve the previous results.