Papers

In [Elkies 2006, Elkies 2007] we announced a K3 surface X/Q with Néron–Severi group NS(X) = NS_Q(X) of rank 19 and an elliptic fibration of Mordell–Weil rank 17. This is the largest possible Mordell–Weil rank over Q(t) for an elliptic K3 surface. One of the fibers of this surface is the elliptic curve of rank at least 28 that was the elliptic curve of highest rank known during the years 2006–2024. We further announced that there are quadratic base changes of this fibration to elliptic surfaces of Mordell–Weil rank 18 over Q(t), and that there are pairs of such quadratic base changes whose compositum is an elliptic surface of Mordell–Weil rank 19 over an elliptic curve E_0/Q with #(E_0(Q)) = \infty, whence there are infinitely many elliptic curves of rank at least 19 over Q. We exhibit one such pair. A subsequent paper will show (using a technique also announced in [Elkies 2007] how we computed X and its fibration.

The set of $\tau$-approximable numbers, $\mathcal W(\tau)$, has genuinely fractional Hausdorff dimension, whereas the set of inhomogeneously badly approximable numbers, $\Bad^\gamma$, has full Hausdorff dimension. We determine the Hausdorff dimension of their intersection by introducing the $\Psi$-rapid game, a scale-sensitive refinement of the rapid game of Hatefi and Simmons (preprint 2024). For every approximation function $\psi$, we prove that $\mathcal W(\psi)\cap\Bad^\gamma$ is strong $\Psi$-rapid winning for a natural gauge $\Psi$ determined by $\psi$. Unlike Schmidt-type games, whose winning property always implies full Hausdorff dimension, the $\Psi$-rapid game is calibrated to a prescribed Diophantine scale, so that the resulting dimension bound depends explicitly on the decay of $\Psi$. In particular, for $\psi(q)=q^{-\tau}, \tau\ge1,$ we recover the exact Jarník–Besicovitch dimension, that is, $$ \HD\bigl(\mathcal W(\tau)\cap\Bad^\gamma\bigr)=\frac{2}{\tau+1}.$$

We determine the Hausdorff dimension of the intersection of the set $\mathcal W_3(\lambda)$ of simultaneously $\lambda$-well approximable points with the Veronese curve $\mathcal V_3 \subset \mathbb R^3$ for $\lambda \ge3/5$, thus completing the full range of $\lambda$ values. Precisely, we show that for $\lambda\ge 1/3$, $$ \dim\bigl(\mathcal W_3(\lambda)\cap\mathcal V_3\bigr)= \max\left\{\frac{2-2\lambda}{1+\lambda}, \frac{2}{3(1+\lambda)}\right\}. $$ To the best of the authors' knowledge, this makes $\mathcal V_3$ the first nondegenerate, non-planar curve with a completely determined Hausdorff dimension theory.

In this article, we develop vertical control theorems for fine Selmer groups associated with Galois representations and their deformations in $p$-adic families. We apply them to provide new evidence for Conjectures A and B of Coates–Sujatha and to obtain pseudonullity of the fine Selmer groups for cyclotomic characters, elliptic curves, classical cuspidal newforms, and Hida families of $p$-ordinary cuspidal newforms.

For positive integers $q$ and $m$ with $m\geq 2q+3$, consider the symmetric numerical semigroup $S_{m,q}=\langle m,m+1,qm+2q+2,qm+2q+3,\ldots,qm+m-1\rangle$. Ciolan, Garc'ia-S'anchez, and Moree asked whether every member of this family with embedding dimension at least $4$ is noncyclotomic. We answer this question affirmatively for every $q\geq 1$, including an independent proof of the previously known case $q=1$. Let $P_{m,q}$ be the semigroup polynomial, and set $t=m-2q-3$, $L=2(q+1)(m+1)-1$, and $D=\deg P_{m,q}=2qm+2q+2$. For $m\geq 2q+4$, let $\rho_1,\ldots,\rho_D$ be the roots of $P_{m,q}$, counted with multiplicity. An explicit computation of a single coefficient of the formal logarithm gives $\left|\sum_{j=1}^{D}\rho_j^{-L}\right|=L\left|[x^L]\log P_{m,q}(x)\right|\geq Lt-1>D$. The strict inequality forces $P_{m,q}$ to have a root off the unit circle. Hence $S_{m,q}$ is noncyclotomic whenever $m\geq 2q+4$. The boundary case $m=2q+3$ has embedding dimension $3$ and is cyclotomic. Therefore $S_{m,q}$ is cyclotomic if and only if $m=2q+3$.

We introduce a matrix divisor function $\tau_n(T,M)$, counting factorisations $AB=M$ for $n\times n$ integer matrices $A,B$ of height at most $T$. For a fixed non-singular $M$, or for the zero matrix $M=O_n$, we prove an asymptotic formula for $\tau_n(T,M)$, as $T\to \infty$, using lattice point counting. We also prove an essentially sharp uniform upper bound for $\tau_n(T,M)$, for an arbitrary non-singular matrix $M$.

The uniform Littlewood conjecture (ULC), introduced by Bandi, Fregoli and Kleinbock, asserts in the two-number case that $$ \lim_{Q\to\infty} Q\min_{1\le n\le Q}\|n\xi\|\,\|n\zeta\|=0 $$ for all real $\xi,\zeta$. It is proven to hold for almost every pair $(\xi,\zeta)$. Schleischitz, however, has recently disproved the full statement and showed that the set of counterexamples contains a dense $G_\delta$ set. We prove that a set of counterexample pairs with the first coordinate being a badly approximable number has Hausdorff dimension at least $3/2$. We further show that the set of badly approximable numbers $\xi$ for which there exists $\zeta$ such that $(\xi,\zeta)$ is a counterexample to ULC has full Hausdorff dimension. This contrasts with the classical Littlewood conjecture, for which the set of possible counterexamples is known to have Hausdorff dimension $0$.

We prove three results about refined $SU(N)$ VW invariants on Hirzebruch surfaces. First, we crucially simplify the modular anomaly equation satisfied by the generating series of these invariants by expressing it in terms of complementary generalized error functions, which requires extending these error functions to their singular loci. Second, we evaluate the variation of the modular completions of the generating series upon change of polarization. Finally, a representation of the generating functions in terms of indefinite theta series allows us to compute VW invariants at the walls of marginal stability, and to express them in terms of the invariants on the two sides of the wall and those of lower ranks. All the results exhibit a universal structure suggesting their wider applicability.

The oscillation of M(x), the partial sum of the Möbius function has been in the focus of researchers in the theory of primes since the famous conjecture of Mertens in 1905 (formulated in a weaker form by Stieltjes in 1885 in a letter to Hermite). The average order of the modulus of M(x) in an interval of type [0,Y] is clearly in connection with the distribution of primes. The author proved at the beginning of 1980's that this average tends unconditionally to infinity with Y. The present work shows that (similarly to the average order of the error term of the Prime Number Theorem) this average agrees with great accuracy for large values of Y with the largest error term of the Riemann-von Mangoldt prime number formula for the value Y.

We present a quaternionic proof that the quadratic form $t^2+2x^2+5y^2+10z^2$ represents all positive integers, and that the form $t^2+x^2+7y^2+7z^2$ represents all natural numbers which are neither equal to $3 \cdot 7^\ell$ nor equal to $6 \cdot 7^\ell$, for any $\ell \in \mathbb{N}_0$. To do this we introduce a technique that may be useful for other purposes, and that we call "metaconjugation".

Let $G$ be a quasi-split reductive group over a finite field and let $\^G$ be the Langlands dual group. Assuming the derived subgroup of $G$ is almost simple, we use techniques from the geometric Langlands program to relate special values of Bessel functions for generic principal series representations of $G$ to the trace of Frobenius acting on Kloosterman sheaves of Heinloth-Ngô-Yun for $\^G$. We also give explicit examples in essentially all possible cases, including exceptional groups.

Weil's criterion states that the Riemann Hypothesis (RH) is equivalent to the non-negativity of an explicit quadratic form Q(f) on test functions. For f supported in [-L,L] we study the normalized infimum lambda_min(L) = inf Q(f)/||f||^2 from both sides. Lower bounds (unconditional): positivity on bounded support is classical for supp f in [-(log 2)/2,(log 2)/2] (Yoshida; Connes-Consani). We prove by certified computation that Q(f) >= 8.9e-18 ||f||^2 for all supp f in [-0.8,0.8], i.e. autocorrelation support 1.6, 2.3 times the classical range. The proof rests on a one-stroke reduction: a pointwise envelope for the Weil symbol, with a comb constant shown optimal by Weyl equidistribution, converts positivity on the whole window into positive semidefiniteness of a single finite matrix. The odd parity sector shows that the certified positivity holds for arbitrary complex test functions of support 1.6, and that the window ground state is simple and even, the spectral hypothesis required by the operator-theoretic program of Connes, Consani, Moscovici and van Suijlekom. Upper bounds (unconditional): variational bounds are evaluated on the geometric side in interval arithmetic, with no appeal to zeros or RH, down to 3.2e-283 at L=2. They follow the empirical law -ln lambda_min(L) ~ 2 pi^2 N(T*)/ln N(T*), T* = 2 pi e^{2L}, with N the zero-counting function; 2 pi^2 matches the Landau-Widom eigenvalue-plunge rate. Under RH, lambda_min(L) <= exp(-L e^L) for all large L. Synthesis: at L=0.8 the two halves enclose the profile, 8.9e-18 <= lambda_min(0.8) <= 2.27e-17, both certified. We show why the positivity route cannot reach RH unassisted: any one-stroke certificate must resolve frequencies up to 2 pi e^{A_L}, A_L ~ 4e^L, a doubly exponential threshold that no pointwise bound on the prime comb lowers, while the spectral margin collapses at the Landau-Widom rate.

Jungin Lee in 2018 proved a necessary and sufficient condition that an integral quadratic form $\sum_{i=1}^{m} a_iX_i^2$ is universal over $M_2(\mathbb{Z})$. For a positive integer $n \geq 2$, Lee defined $f(n)$ to be the smallest positive integer $m$ such that for every pairwise coprime $a_1, a_2, \ldots a_m \in \mathbb{Z}$, $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z})$. He gave bounds on $f(n)$ too. Koo and Lee further improved the bounds in 2025. This paper is organized as follows. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. In the first section we give necessary and sufficient condition for a quadratic form $\sum_{i=1}^{m}a_iX_i^2$ to be universal over $M_2(\mathbb{Z}_p)$ for $p=2$ and for an odd prime $p$. Consequently we express matrices of over $\mathbb{Z}_p$ as sum of squares. For a positive integer $n \geq 3$, we define $f(n)$ to be the smallest positive integer $m$ such that for every $p$-adic integers $a_1,a_2, \ldots a_m$ with at least three of them units, the diagonal quadratic form $\sum_{i=1}^{m}a_iX_i^2$ is universal over $M_n(\mathbb{Z}_p)$. In the next section we find bounds on $f(n)$ for $n \geq 3$.

We investigate the distribution of the zeros of the Euler's function and its integral and fractional derivatives.

We develop the structure theory of the deterministic $\sin^2$-type algorithm for complex cubic fields introduced in the companion paper, addressing the complex-signature case of Karpenkov's Problem 4. The selection rule is shown to be, exactly, the minimization of a conformal module: the hyperbolic cosine of the distance between the transverse complex structure of the state and the round point. All governing quantities are exact elements of the real embedding of the field and satisfy closed dual-type identities; in particular no isotropic candidate ever arises, and the transverse deviation lattice has exactly pinned covolume. We prove an unconditional soft-rebound lemma (the module can grow by at most the factor $\varphi^2 = 2.618\ldots$ in one step), a finiteness theorem for states of bounded module and height at fixed coordinate discriminant, with explicit static constants, and a per-field periodicity theorem under two named hypotheses: $(C_\kappa)$, contraction of the module in the high phase, partially reduced here to a fixed finite minimax over a five-parameter compact with rational objective; and (B), recurrence of bounded height, which we then prove under $(C_\kappa)$ alone: a height-descent theorem shows the height can never exceed $\max(H(s_0), C_H)$ with an explicit constant. The remaining program for per-field periodicity is reduced to (R) on the compact and to the proved stretched subcases. All proved statements and certificates are finite and exact. A machine-checked core of the paper is sealed in Lean 4, kernel-only, under the standard axioms: the analytic core of the height-descent theorem, the finiteness pigeonhole, the dual and conformal identity layer, and an abstract assembly theorem composing them through named interface hypotheses.

Let $k=\mathbb F_q$. We reconstruct every $k$-indecomposable $g\in k(X)$ of degree greater than one whose normal closure has genus one. Such a map is automatically separable and, after independent degree-one changes of the source and target coordinates over $k$, arises from a separable equivariant isogeny between elliptic curves equipped with compatible finite group actions stable under Frobenius conjugation. In particular, $\deg g=\ell$ or $\ell^2$ for a prime $\ell$, with $\ell\ne\operatorname{char}k$ in the latter case. More generally, every separable rational function with genus-one Galois closure admits a canonical factorization class, modulo degree-one changes of the intermediate coordinates over $k$, determined by the intrinsic translation subgroup of its geometric monodromy group; every $k$-indecomposable factor of the remaining map has Galois closure of genus zero. For every equivariant-isogeny quotient and every finite extension $k_r/k$, the same exact finite-kernel condition characterizes both permutation of $\mathbf P^1(k_r)$ and exceptionality over $k_r$. The finite kernel and its induced Frobenius and linear symmetry actions also determine the arithmetic and geometric monodromy permutation groups, decomposition classes, and the exact periodic set of permutation extension degrees, including its least period and limiting proportion. Together with the corresponding genus-zero theorem, these results yield permutation if and only if exceptionality over every finite extension for every separable rational function whose Galois closure has genus at most one; under $k$-decomposition, the common extension-degree set is the intersection of the corresponding sets for the factors.

Every number field canonically gives rise to two lattices: its ring of integers and its log unit lattice. While the shapes of the former have undergone extensive research, far less is known about the shapes of the latter, referred to as unit shapes. This paper presents an in-depth analysis of the unit shapes of number fields with unit rank 2. Our first main result characterizes, in many cases, the location of unit shapes within the fundamental domain of the space of rank 2 lattice shapes in terms of the Galois group of the field's Galois closure, and determines when these unit shapes are transcendental. Next, we establish that the unit shape uniquely determines the field up to isomorphism for totally imaginary $D_6$ non-CM sextic fields; this result fails in the CM case. Finally, for certain subfamilies of $D_6$ non-CM imaginary sextics, we offer a simple sufficient condition for their log unit lattices to be orthogonal and provide lower bounds on the proportion of fields with orthogonal log unit lattice.

We study degenerate Eisenstein series on symplectic groups and construct a new automorphic descent. We show that a certain Fourier coefficient, after restriction to a smaller symplectic group, is itself an Eisenstein series with the same inducing parameter \(s\). We describe the resulting holomorphic section explicitly in terms of the original section and local \(L\)-factors at unramified places. At every regular point of the local descent, we prove surjectivity at non-Archimedean places and dense image at Archimedean places. We finally indicate an application of the iterated descent to maximal Fourier coefficients of Eisenstein series.

Let $k=\mathbb{F}_q$. A polynomial representative of a finite-set map is a tuple of polynomials inducing that map on the rational-point grid. We prove a sharp distinction between a finite-set map and the geometry of its representatives. If $n=1$ or $n=2$, every polynomial representative of a permutation of $k^n$ has algebraically independent coordinates. If $n\geq3$, every set map $k^n\to k^n$ has both an algebraically independent and an algebraically dependent representative; the latter may be chosen to satisfy \[ F_2^q-F_2=(F_1^q-F_1)F_3. \] More generally, every map $k^m\to k^n$ has an algebraically independent representative exactly when $n\leq m$, while every such map has a dependent representative when $n\geq3$. The dependent construction combines an Artin–Schreier interpolation theorem, producing prescribed values by polynomials $A,B$ with $A^q-A\mid B^q-B$, with a three-coordinate suspension. For the identity on $k^3$, the scheme-theoretic image may be chosen to be exactly \[ V^q-V=(U^q-U)W, \] a smooth geometrically integral rational surface. We also establish low-degree and extension-field criteria forcing algebraic independence. An exact exhaustive computation additionally proves that every $2$-reduced representative of a permutation of $\mathbb{F}_2^3$ has algebraically independent coordinates.

We study second-order asymptotics for optimal thresholds in secretary-type problems. Under a unimodality assumption and a local affine expansion of the discrete payoff increments, we establish a general principle showing that sufficiently accurate rational approximations to the leading threshold proportion yield exact optimal integer thresholds. Continued-fraction convergents provide an immediate application. We also derive an asymptotic rounding law and apply the results to several variants of the secretary problem with fixed and random horizons.

The paper investigates the topological and metric properties of the set $E$ of values of all infinite continued fractions of the form \[0+\frac{1}{a_1+\dfrac{1}{a_2+_{\ddots}}}=1/a_1+1/a_2+...+1/a_n+...\equiv[0;a_1,a_2,...,a_n,...],\] whose partial quotients ($a_n$) take values in a finite set of positive real numbers $\{e_0,e_1,...,e_{s-1}\}$, $e_0<e_1<...<e_{s-1}$. It is established that the set $E$ is bounded, has cardinality continuum, and is perfect. Conditions under which $E$ is an interval, a nowhere dense set, or a set of Lebesgue measure zero are established. Necessary and sufficient conditions are obtained for $E$ to be an interval and for the corresponding coding system by such continued fractions to have zero redundancy, i.e., for each number to have at most two representations. The geometric meaning of the digits in this representation, as well as the metric relations, is described in terms of the properties of cylindrical sets (cylinders and cylindrical intervals). It is proved that the basic metric ratio, defined as the ratio of the diameter of a cylinder to the diameter of its parent cylinder, is bounded away from both zero and one.

Let $p,q,r\geq 2$ and consider the Campana orbifold \[ \left( \mathbb{P}^1, \left(1-\tfrac1p\right)[0] +\left(1-\tfrac1q\right)[1] +\left(1-\tfrac1r\right)[\infty] \right). \] Primitive positive Campana points on this orbifold correspond to solutions of $a+b=c$ in which $a$, $b$, and $c$ are respectively $p$-full, $q$-full, and $r$-full. We establish upper bounds for the number of such points of bounded height in a broad range of exponents, with a power-saving over the trivial bound. The main analytic input is an estimate for primitive integral points in lopsided boxes on generalized Fermat surfaces $a_1x^p+a_2y^q+a_3z^r=0$, which is uniform in the coefficients.

We provide an explicit bound for the image of the Zilber–Pink locus of a curve in an abelian variety under reduction modulo a suitably large prime $p$. We also provide a new proof of finiteness of the Zilber–Pink locus. Under further assumptions on the abelian variety, we prove a bound on the size of the locus.

In this paper, we prove that the set of counterexamples to the uniform Littlewood's conjecture proposed in Bandi-Fregoli-Kleinbock, that is, the set of pairs of real numbers $(x,y)$ satisfying $$ \limsup_{Q\to +\infty}\ Q\cdot \min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs $(x,y)$ satisfying $$ \liminf_{m\to +\infty}\ Q_m\cdot \min_{1\leq q\leq Q_m}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning if $Q_{m+1} \gg Q_m^{\tau}$ for some $\tau > 1$. In particular, the above sets have full Hausdorff dimension in $\mathbb{R}^2$. In addition, we prove that these sets are absolute winning on every regular $C^2$ planar curve whose set of points of nonzero curvature is itself absolute winning on the curve. We also establish analogous results for certain lines.

A decade ago, locally harmonic Maaß forms were first established by Bringmann, Kane, and Kohnen in negative weights and independently by Hövel in weight $0$. Since then, they have seen important applications related to twisted central $L$-values of newforms. However, there are just a few examples of these forms known so far, and in particular there is no unifying theory behind these examples. Towards this direction, we show that the representations of Zagier's $f_{k,D}$ and Mono's $g_{k+1,D}$ functions as traces of cycle integrals are not unique.

Let $C/K$ be a curve of genus 2 over an algebraically closed field $K$. Every such curve comes equipped with a canonical quadratic form $q_C$ called its refined Humbert invariant. In the case that the Jacobian $J_C$ of $C$ is isogenous to the self-product $E \times E$ for an elliptic curve $E/K$ with complex multiplication (CM), we provide an explicit formula for the finite number $N_C$ of isomorphism classes of genus $2$ curves $C'/K$ whose refined Humbert invariant is equivalent to $q_C$. This formula implies that $N_C$ is unbounded for such curves, and that there are only finitely many isomorphism classes of such curves $C/K$ with a given value of $N_C$. A key step in our approach is a characterization of when $J_C$ is isogenous to a self product of a CM elliptic curve, formulated purely in terms of properties of the refined Humbert invariant $q_C$. We establish that an analogous characterization also holds for superspecial curves of genus 2. The paper concludes with explicit examples illustrating cases where a genus 2 curve is uniquely determined by the invariant $q_C$.

For primes $n\equiv1\pmod{24}$ we study how deep an explicit, factorization-free search for a decomposition of $4/n$ into three unit fractions has to go. Write $E_2(N;J)$ for the set of such primes $n\le N$ at which no witness of depth at most $J$ exists, in the sense of the two-parameter criterion of Theorem 3.9 with coprime parameters $u,a\le J$. We prove that for every fixed $J$ $$|E_2(N;J)| \ll_J \frac{N}{(\log N)^{1+\mathfrak{A}(J)/2}},$$ the exponent being the exact dimension of the covering on which the proof rests. The proof replaces the subgroup generated by the prime factors, an approach that breaks down as soon as $(\mathbb{Z}/4m)^{\times}$ has exponent greater than $2$, by a fixed-point-free involution, and is unconditional at every $J$. Second, we exhibit an unconditional obstruction. At the shift $c=7$ there are $\asymp N(\log N)^{-3/2}$ primes $n\le N$, $n\equiv1\pmod{24}$, at which both branches of the divisor criterion fail. The representation of $K_7=(n+7)/4$ by the principal form of discriminant $-7$ has to be primitive, so that the relevant input is the primitive-representation theorem of Fuchs, Hsu, Rickards, Schindler and Stange [25] rather than the classical results of Iwaniec; a fixed shift therefore cannot leave a finite residual set. Third, a factorization-free procedure decides the conjecture for all primes of an interval $[N,2N]$. Its Type II pass costs $\mathcal{O}(N(\log N)^{3})$ while its Type I pass costs $\Theta(N^{2})$, which locates the whole quadratic cost in the extraction of the divisors of $4u^{2}d+1$ and exhibits an asymmetry between the two halves of the Type I/Type II dichotomy. The conjecture itself remains open.

For a smooth projective curve over a finite field with a fixed point at infinity, we establish a universal correspondence between finite residue duality and geometric kernel functions. We prove that the finite residue Casimir tensor is realized geometrically as the intrinsic principal part of the normalized Szegő kernel for any acyclic line bundle, and equivalently that this kernel acts as a reproducing kernel for the finite residue pairing, in exact analogy with the classical Cauchy integral formula. In the polynomial case these equivalent descriptions yield a closed formula involving the rank-two Weil operator, identified with the classical divided difference, recovering a remainder identity of Hu–Ou. These results provide a geometric foundation for the study of Anderson generating functions and the Weil pairing for Drinfeld modules.

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $\rho$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-\rho} \] with success probability $1-\delta$. Provided that $\rho$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/\delta))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in\mathbb{N}$, and let $\mu_t$ be the probability measure on $\mathbb{Z}$ defined by the asymptotic densities of the level sets of the function $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in\mathbb{Z}$. Let $P_t$ be the family of finitely supported measures defined by the convolution $\mu_t=\mu_1*P_t$. Recently, Tarlowski (2026) has shown that the family $P_t$ may be represented as a recursively grown binary tree $T_t$, and that the Cusick's conjecture - $\mu_t(\mathbb{N})>\frac12$, $t\in\mathbb{N}$, - follows from the asymmetry property of the family $T_t$, which was posed there as an open problem. Next, Cheng (2026) has provided the combinatorial description of the family $T_t$ in the language of principal subsequence ideals, and proved both conjectures. Both of these problems are directly related to the problem of determining the zeros of the function $\mathbb{N}\ni t \mapsto P_t(\mathbb{N})-\frac12\in[0,\tfrac12]$, a problem left open by Cheng (2026) as a saturation problem, and previously analyzed only numerically. In this paper we solve this problem completely. Writing an odd integer $t\ge3$ as $t=(1\,w\,1)_2$ with $w\in\{0,1\}^{\star}$, we show that $P_t(\mathbb{N})=\frac12$ if and only if $w$ is saturated in the following sense: in the block decomposition $w=1^{a_0}\,0\,1^{a_1}\,0\cdots0\,1^{a_k}$ with exactly $k$ zeros, every block of "1" satisfies $a_i\ge k$. Additionally, we show that the lower bound for $P_t(\mathbb{N})$ established by Cheng for $0$-initial words holds true for all non-saturated words.