Papers

Let $p$ be an odd prime, let $m\geq1$ be an integer, and let $Q$ be a nondegenerate quadratic form on $\mathbb F_p^{2m}$. For a nonempty set $E\subseteq\mathbb F_p^{2m}$, write $\Delta_Q(E)=\{Q(x-y):x,y\in E\}$. We prove that, whenever $|E|>p^m$, \[ |\Delta_Q(E)|\gg \min\left\{\frac{p}{L_E},\, \frac{|E|}{p^{m-1}L_E^2}\right\}, \qquad L_E=\log\left(2+\frac{p^{m+1}}{|E|}\right), \] with an absolute implied constant independent of $p$, $m$ and $Q$. This confirms the Erdős–Falconer distance conjecture up to a logarithmic factor. In the anisotropic planar case $m=1$, for small sets, we also prove that \[ |\Delta_Q(E)|\gg\frac{|E|}{\log(2|E|)}, \qquad (1\leq |E|\leq p), \] which is also optimal up to a logarithmic factor.

We study three recurrent double sequences whose colored diagrams encode mirrored versions of the finite-ordinal construction. Two sequences use three letters and realize variants with two and three initial symbols. A four-letter sequence realizes the von Neumann construction from one initial symbol. For each sequence we describe its cells explicitly and identify the ordered nesting of its rectangles. For four letters, we distinguish ordinary and virtual boxes and prove their assembly rules. The proofs proceed by induction along antidiagonals. For the constructions with one and three initial symbols, we reduce the recurrence to finite sets of local configurations and verify those sets by computer. We derive formulas for the numbers of nested boxes, including arithmetic-term representations, and we prove that none of the three sequences is $k$-automatic for any integer $k \geq 2$.

In this paper, we study the number of points of $\mathbb{Z}$-schemes $X$ of finite type over the finite rings $\mathbb{Z}/p^m\mathbb{Z}$ for positive integers $m$ and prime numbers $p$. Motivated by Igusa's conjecture for exponential sums and the Lang-Weil estimate, we establish an estimate of $X(\mathbb{Z}/p^m\mathbb{Z})$ in terms of $m$, $X(\mathbb{Z}/p^{m-1}\mathbb{Z})$ and poles of the $p$-adic Igusa local zeta function of $X$ uniformly in $m\geq 2$ and $p$. Our estimate might be viewed as an inductive version for counting points of schemes over finite rings of the Lang-Weil estimate. Our estimate agrees with the expected bound of Igusa's conjecture for exponential sums if $X_{\mathbb{Q}}$ is not a locally complete intersection having only rational singularities. This can be regarded as an analogous version of the estimate of exponential sums modulo powers of primes of polynomials with non-rational singularities given by Cluckers, Mustaţǎ and the author in 2019. In the case that $X_{\mathbb{Q}}$ is a locally complete intersection having only rational singularities, our estimate is very close to the expected bound of Igusa's conjecture for exponential sums. To prove our estimate, beside using the usual tools such as resolutions of singularities and Denef's formula for Igusa local zeta functions, we also use the Hermit interpolation formula to study the numerators of Igusa local zeta functions. By the same method, we are able to approach Igusa's conjecture for exponential sums in the case of rational singularities. Lastly, to have further applications of our result in the future, we study heavily on $p$-adic analysis to give an upper bound of the real parts of non-trivial poles of Igusa local zeta functions in terms of suitable generalized log canonical thresholds.

Two integer partitions $t$-intersect if they have at least $t$ common parts, counted with multiplicity. We study the largest $t$-intersecting families of integer partitions of $n$ into exactly $k$ positive parts. The canonical $t$-star consists of the partitions containing at least $t$ ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever $n\ge Ak^3$, for every fixed $A>24$ and all sufficiently large $k$, uniformly over $1\le t<k$. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large $k$ and for every $1\le d<k$, one may choose $d/4\le t\le d$ and $n=\lfloor tk^2/3\rfloor$ so that a $t$-intersecting family is strictly larger than the canonical star.

Let $p$ be a large prime and $\chi$ a non-trivial Dirichlet character modulo $p$. We study the character sum \[ \sum_{a \sim A} \sum_{b \sim B} \sum_{c \sim C} \alpha(a,b) \beta(c)\chi(f(a) + bc), \] where $z \sim Z$ means $Z\le z < 2Z$, $f \in \mathbb{Z}[X]$ is of small degree $k$, and $\boldsymbol\alpha=(\alpha(a,b))_{a\sim A,b\sim B}$ and $\boldsymbol\beta=(\beta(c))_{c\sim C}$ are two complex coefficients. We prove non-trivial upper bounds for this sum in either of the two cases: (1) $\boldsymbol\beta\equiv1$, $k=2,3,4,5$ and $A,B,C>p^{\frac{1}{8}+\varepsilon}$, (2) $\boldsymbol\beta$ general, $k=2,3$ and $A,B,C>p^{\frac{1}{6}+\varepsilon}$, where $\varepsilon>0$ is fixed. This work was originally motivated by an intermediate result of Ganguly and Rajan (2023) on counting $2\times2$ matrices over $\mathbb{F}_p$ with irreducible characteristic polynomials, where the entries are in short segments. The new bounds here allow us to count such matrices in much shorter segments.

Let $a_t(n)$ denote the coefficients of the Nekrasov–Okounkov hook length generating function $F_{t^2}(x)$, and let $b_t(n)$ denote the coefficients of the $t$-core partition generating function $C_t(x)$. We prove a sharp prime-power dichotomy: for every integer $t\ge 2$, the congruence $$ a_t(n)\equiv b_t(n)\pmod t $$ holds for all $n\ge 0$ if and only if $t$ is a prime power. When $t=p$ is prime, the congruence strengthens to modulo $p^2$. For $t=6$, and more generally for every integer that is not a prime power, the congruence fails. This identifies prime powers as the exact moduli for which these two partition-theoretic coefficient sequences are arithmetically equivalent.

For integers $m\ge 2$ and rows $N>m$ divisible by $m$, consider the restricted binomial greatest common divisor $G(N;m)=\gcd\{\binom Nk:0<k<N,\ m\mid k\}$. Fix a prime $p$ with $p\nmid m$, and let $r_p(m)$ be the least positive integer $r$ such that $m<p^r$. We prove that the largest possible value of $v_p(G(N;m))$, as $N$ ranges over all admissible rows, is exactly $r_p(m)$, and we give a constructive equality row. Attainment is established before the least extremal row $T_p(m)$ is defined. For the special family $m=p^a+1$ with $a\ge2$, we determine that least row exactly: $T_p(p^a+1)=p^{3a}+1$. The proof combines Kummer's carry theorem with an explicit leading-digit witness for the universal upper bound, a multiplicative-order construction for equality, and a separate strict-minimality argument below $p^{3a}+1$ with a fallback witness for the unique worst leading-digit pattern. The selected-GCD family itself is known in the literature; the relation to earlier results and the limits of the documented literature search are stated explicitly.

Let $Q$ be a translation invariant indefinite quadratic form in $8$ variables. Let $\mathcal{P}$ be the set of all primes. Let $\mathcal{A}\subset\mathcal{P}\cap\{1,\dots,X\}$. By developing the recent work of Green, we prove that subject to a mild condition on the form $Q$ there exist pairwise distinct primes $p_{1},\dots,p_{8}\in \mathcal{A}$ satisfying $Q(p_{1},\dots,p_{8})=0$ provided that \[\frac{|\mathcal{A}|}{X/\log X}\gg_{Q}(\log\log X)^{-c}\] for some absolute constant $c>0$.

We prove new large value estimates for the Riemann zeta function on the critical line. For instance, we improve the upper bound on the measure of $t\in [T, 2T]$ with $|\zeta(1/2 + it)| \geq T^{1/8}$ for the first time since Hardy-Littlewood (1923). Our results imply improved upper bounds on the $k$-th moment of zeta for every $4 < k \leq 12$. We show in particular that \begin{equation*} \int_{0}^{T} |\zeta(\tfrac{1}{2} + it)|^6 \,d t \ll_\varepsilon T^{\frac{5}{4} - \frac{1}{60} + \varepsilon}. \end{equation*} The main ingredient is a new large value estimate for exponential sums with square-root phases, and certain perturbations. Those arise in the expansion of the short second moment of zeta.

We construct indefinite theta representations for a family of signed generalized Nahm sums of arbitrary rank \(r\geq2\). After explicit normalization and boundary correction, one signed branch gives the even components of a \((2r-1)\)-component holomorphic theta vector associated with a lattice of signature \((1,1)\). Euler-type multiple sums provide the odd components, yielding formulas uniform in both the rank and the component index. Using Zwegers' theory, we determine the weight-one modular transformation laws of the real-analytic completion, with an explicit finite sine matrix for the \(S\)-transformation. We decompose the nonholomorphic correction into a finite sum of products of unary theta functions and one-dimensional error-function series, and compute the antiholomorphic derivative of the completion. Together with the completed transformation laws, these formulas give the inhomogeneous transformation law of the holomorphic vector. We also evaluate a linear combination of two signed branches as a difference of Jacobi product quotients. This identity and further evaluations for two families of symmetrizable matrices and their duals, combined with known results, give multiple-sum representations for every component of four associated product vectors. These representations are finite combinations of generalized Nahm sums with the prescribed matrices and symmetrizers; in the dual families, half-period translates are absorbed into signs and constant phase factors.

In this paper, we construct an explicit correspondence between Engel and Pierce expansion digit sequences, obtained by adding $n-2$ to the $n$th Engel expansion digit, and show that it induces a homeomorphism from the set of irrationals in the unit interval onto the complement of a countable dense subset of this set. This yields a transfer principle for Baire category between the two expansions, and we determine exactly which Borel classes are preserved. We then identify this homeomorphism with the composition of two maps considered by Moroz (2027), and study the continuity of the second map, which sends modified Engel expansions to Pierce expansions. We determine its set of discontinuities, show that every discontinuity is a jump, and prove that the map is nowhere monotone. In particular, this confirms, for this map, the nowhere monotonicity and the continuity outside the exceptional set conjectured by Moroz (2027). As applications, we characterize the non-increasing position-dependent weights for which the associated series of Pierce expansion digits diverges on a comeager set, and we relate the convergence exponents of the Engel and Pierce expansion digit sequences.

A magic square of order $n$ is a square array of $n^2$ distinct integers such that the integers of every row, every column and the two diagonals have the same common sum. If a magic square remains a magic square even when each integer is replaced by its square, it is called bimagic. If, however, all the rows and all the columns of the aforementioned square of squares add up to a common sum, but the two diagonals do not, the original magic square is called pseudo-bimagic. Except for one family of $6 \times 6$ pseudo-bimagic squares, parametrized families of bimagic squares have not been published. In this paper we construct parametrized families of $6 \times 6$ pseudo-bimagic squares and $8 \times 8$ bimagic squares. We use these families to construct, for any positive integer $m$, multi-parameter families of pseudo-bimagic squares of order $6m$ and bimagic squares of order $8m$. We also give numerical examples of $6 \times 6$ and $12 \times 12$ pseudo-bimagic squares and an $8 \times 8$ bimagic square.

In this paper, we study the presence of perfect powers in orbits of polynomials, or of semigroups generated by a finite family of polynomials, defined over a number field. In particular, we correct and generalize a result of Ahmadi et al. (2012), showing that multiplicative shifts of perfect powers in orbits can be detected by a finite algorithm. We also give a finiteness result on elements in orbits satisfying certain multiplicative relations.

We prove the transcendence of numbers that have $\beta$-expansions in algebraic bases $\beta$ belonging to dendric shifts with purely discrete spectrum, under certain combinatorial conditions on their directive sequences of substitutions. This includes Arnoux-Rauzy sequences on arbitrary alphabets and Cassaigne-Selmer sequences. The proof uses Rauzy fractals and the subspace theorem.

We study the one-dimensional Dirichlet spectrum and obtain new results concerning the origin $\mu$ of the ray contained in the spectrum and its isolated points. First, we prove the improved upper bound $\mu \leq (448875+1611\sqrt{67773})/46982 = 18.4809\ldots$. We also exhibit an explicit gap $(L,R)$ in the Dirichlet spectrum, which yields the improved lower bound $\mu \geq R = 5.9596\ldots$. The proof of the lower bound includes an exact computer-assisted finite search over admissible continued-fraction words. We further prove that all points $Z_k$, $k\geq 2$, introduced by Ivanov are isolated points of the Dirichlet spectrum. Finally, we determine the asymptotic behavior of the neighboring gaps around $Z_k$ and describe the equivalence classes of all irrational numbers whose Dirichlet value is $Z_k$.

Adapting the circle method developed by Frantzikinakis–Klurman–Moreira, we establish a sufficient criterion for a set of natural numbers to be multiplicatively recurrent. We apply this criterion to Piatetski–Shapiro sets with small exponents and to regular irrational nil-Bohr sets. We also study the value distribution of unimodular completely multiplicative functions along these sets.

Let $A$ and $B$ be polynomials with integer coefficients. We study nonzero integer solutions of $$y\mid A(x),\qquad x\mid B(y),$$ using quotient transformations that may change the polynomial pair. We call this process congruence jumping. We establish a general reciprocal-polynomial rule governing such jumps, without coprimality, monicity, or unit assumptions, and give exact criteria for when the resulting polynomial states can be normalized integrally. Companion-surface identities provide a mechanism for constructing infinite integral quotient chains, including an explicit mixed-degree example. The quadratic case is considerably more rigid. We obtain a quantitative denominator bound for finite chains, classify the exceptional nonconstant one-sided infinite chains with nonintegral conic parameter, and show that changing quadratic states reduce to ordinary Vieta dynamics on a fixed conic. We further construct a genuinely nonunit recurrent four-cycle with infinitely many positive integral points and show that its dynamics admits a uniform Pell-type linearization. A motivating nonunit divisibility system is analyzed through Pell orbits and changing-state ladders. Finally, an independent relation-lattice criterion reduces certain solution sets to a finite divisor search.

Let $k$ be a positive integer, and consider the sequences of positive rationals with $x_0\in\N$ and $x_{n+1}=k/(x_0+x_1+\dots+x_n)$. Write $x_n=a_n/b_n$ in lowest terms. We show that there is a rational constant $c>0$ such that $c\,a_n+b_n$ is a perfect power for every such sequence and every $n\ge2$ if and only if $k=h^2$ and every prime factor of $h$ is congruent to $3$ modulo $4$; in that case $c=2/h$ and the powers are squares. The proof rests on the observation that $t_n=(x_0+\dots+x_n)/h$ satisfies $t_{n+1}=t_n+1/t_n$, a recursion under which reduced fractions never cancel. This yields an exact formula for $c\,a_n+b_n$, shows that along any single sequence each prime factor of $h$ spoils at most one term, and leads to two generalizations. For arbitrary $k$ the invariant $b_n^2-\frac4k a_n^2$ is always a rational square, and it is always an integer square exactly when the primes dividing the square part of $k$ satisfy an inertness condition in $\Q(\sqrt{-k})$. For the recursions $x_{n+1}=h^2/(x_0+\dots+x_n+n\mu h)$ the role of the Gaussian integers is played by the quadratic fields $\Q(\sqrt{\mu^2-4})$, which include $\Q(\sqrt{-3})$ and every real quadratic field.

In this paper, we give exact combinatorial expressions for all CUE joint moments on the unit circle, in the case that all exponents are even positive integers, yielding the corresponding asymptotics. We offer two equivalent expressions for these quantities: the first is a sum involving Kostka numbers; the second is related to counting integer matrices (magic squares). As an application of our results, we unify many results in the literature, which were previously disconnected. Moreover, we prove a conjecture of Basor et al. (2018) regarding an irregular conformal block structure.

A longstanding open problem concerning Ramanujan-Sato series is given by determining an expression for the coefficients $c_{11}(n)$ associated with Ramanujan-Sato series of level 11 with a fixed, nested, finite binomial sum for all nonnegative integers $n$, by direct analogy with the known formulas for the coefficients for lower-level Ramanujan-Sato series (and this is clarified and made explicit in our paper). This open problem was given by Cooper et al. in 2015 [J. Math. Anal. Appl.], in Cooper's 2017 textbook on Ramanujan's theta functions, and in Cooper's 2025 contribution to the Contemp. Math. series. This problem seems to have remained open. We succeed in solving this problem, by introducing and proving a double binomial sum formula for $c_{11}(n)$ that is directly analogous to the coefficient formulas for lower-level Ramanujan-Sato series.

The $\hat{Z}$ invariant of a three-manifold $M_{3}$, a half-index of the $3d$ $\mathcal{N}=2$ theory $T[M_{3}]$, is well understood for negative-definite plumbed three-manifolds. For orientation-reversed manifolds, however, the known formulas fail to produce $q$-series, and making sense of the operation $q \rightarrow q^{-1}$ remains a central open problem. Seifert manifolds, whose $\hat{Z}$ invariants are built from false theta functions, offer a natural testing ground. We study “dual false theta functions”, the objects that should replace false theta functions under orientation reversal. Two characterizations of these duals exist in the literature: a modular one, based on mock modularity with a prescribed shadow, and a resurgent one, based on Borel–Mordell integrals and transseries. We prove that the two characterizations are equivalent. We construct a new family of dual false theta functions for values of positive integer $p$ satisfying a Pell-equation condition. Its members are built from Zwegers' indefinite theta functions, have integer coefficients, and have effective central charge $c_{\text{eff}} \leq 1$. We also show that the mock theta functions of Li and Schwagenscheidt satisfy these characterizations and arise from a natural regularization of the divergent product $\eta^{3}(\tau)\psi_{p, a}(- \tau)$.

The integer Chebyshev constant of [0,1] has been bracketed by 0.4213 <= t_Z([0,1]) <= 0.422685 since the work of Pritsker (2005) and of Flammang (2014). We prove 0.4222286000 <= t_Z([0,1]) <= 0.4226846976, narrowing the interval from 1.39e-3 to 4.56e-4. Both bounds are established by explicit certificates. The upper bound is carried by an integer polynomial with 92 factors and explicit integer exponents; the lower bound by a discrete measure on 947 cells together with a library of 206 irreducible polynomials, through an inequality in which the terms paying for divisibility are made explicit. Two self-contained scripts, included as ancillary files, re-derive both bounds from the certificates alone.

Grishkov and Logachev have asked (Problem 1.4 of their recent survey) whether, for a fixed finite field F_q, the analytic ranks of the twists of the Carlitz module are bounded. We answer this in the negative for every q >= 3. For q = p^s and n >= 1 put P_{q,n} = (theta^(q^n) - theta)^(q-2). We show that the directed graph of the matrix M(P_{q,n},1,k) of Grishkov-Logachev has a completely explicit set of circuits: they are pairwise disjoint, none of them meets an entry involving t, and they are in bijection with the orbits of E^n under cyclic rotation, where E is the set of l in [0,q-2] with p not dividing l+1. Consequently L(C_P, t, T) does not depend on t, it factors as a product of one term 1 - wt(nu) T^per(nu) per orbit, and the analytic rank equals the sum of p^(v_p(per(nu))) over the orbits of weight 1. When n is a power of p this equals (q/p)^n (p-1)^(n-1), which is unbounded. The argument also answers questions 3.6.1 to 3.6.4 of the survey for this family.

Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of ordinary reduction for $E$, and assume that $E$ admits a rational $p$-isogeny. In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa $\mu$-invariants of the Selmer groups attached to $E$ over the cyclotomic $\mathbb{Z}_p$-extension of the rational numbers. Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens's conjecture on the behavior of analytic $\mu$-invariants.

We develop a resurgent transseries approach to mock theta functions and show that the associated Mordell–Appell integrals underlie their modular and resurgent structure. From these integrals, we recover the modular transformation laws, obtain uniqueness characterizations of the mock theta vectors, provide a canonical crossing of their natural boundary, and produce new mock theta functions. After representing the Mordell–Appell integrals as Laplace transforms of elementary resurgent functions, we rotate the Laplace contour to a Stokes line. Their Stokes phenomena and their modular transformation laws reproduce those of the associated unary series and, combined with our uniqueness results, those of the mock theta functions. Conversely, we regard these modular transformation laws as functional equations governing the mock theta vectors. For Ramanujan's order-$3$ pair $(f,\omega)$ and the distinguished mock theta vectors of orders $5$, $7$, and $11$, satisfying the full $SL(2,\mathbb Z)$ transformation laws, we prove that these functional equations uniquely determine a holomorphic solution in the unit disk under canonical minimal-growth normalization. This unique solution is precisely the classical mock theta vector. We show that the normalization is essential: for orders $5$, $7$, and $11$ we determine explicit nontrivial solutions of the homogeneous problem, yielding new mock theta functions with faster-growing coefficients. Finally, we give two intrinsic and natural constructions for crossing the natural boundary $|q|=1$. The first continues the modular functional equations and takes their unique holomorphic solutions on the other side. The second acts on the resurgent asymptotic series at the cusp: we replace $\tau$ by $-\tau$ and apply Écalle–Borel summation. We prove that the constructions agree, giving a canonical and explicit boundary crossing.

Hirose formulated a duality conjecture for a $q$-discretization of iterated integrals on the four-punctured projective line, incorporating word-dependent $q$-shifts of the parameters. In this paper, we prove this conjecture using a symmetric terminating ${}_{4}\phi_{3}$ connector.

Quadratic Chabauty computations for $X_0(N)^*$ are complicated by local height contributions at primes of bad reduction. For squarefree $N$, we explain a strategy for constructing a global $p$-adic height for which all the contributions at the primes of bad reduction vanish. As our first main result, we show that such a $p$-adic height exists for all squarefree levels $N>714$. In order to do this, we first give an explicit description of the minimal regular model of $X_0(N)^*$ at primes of bad reduction, and show this model is semi-stable. Each component of this model gives rise to a linear condition which ensures that the contribution at that component vanishes. Using embeddings of quadratic orders into quaternion algebras, we obtain an upper bound for the number of these conditions; when the genus of $X_0(N)^*$ is greater than one more than this bound, there is enough freedom to choose a suitable correspondence, simplifying quadratic Chabauty computations. As an application, we determine the rational points on several curves $X_0(N)^*$ for which this was not done before.

We study random polynomials of the form $R(x)=x^n+\omega_{n-1}x^{n-1}+\cdots+\omega_0$, where $\omega_0,\dots,\omega_{n-1}$ are independent, uniformly bounded integer-valued random variables, and $\omega_1,\dots,\omega_{n-1}$ have a fixed common law $\mu$. We prove (unconditionally) that, if the Rényi entropy of order $2$ satisfies $H_2(\mu)=-\log\|\mu\|_2^2>12$, then $\mathbb{P}(\operatorname{disc}(R)\text{ is a square})=O_\mu(1/\log n)$. Combined with previous results, this shows that, for such measures $\mu$ and under additional hypotheses, $R$ has full Galois group $\mathrm{Sym}(n)$ with high probability, when conditioned on $\omega_0\ne 0$.

This paper is devoted to questions concerning rational quadrilaterals and rational tetrahedra. In both cases there is a finite group which acts on the set. In the case of rational quadrilaterals it was discovered by Kurt Heegner; in the second case there are two variants, one again due to Heegner the other a consequence of an involution introduced by Tullio Regge. These allow one to construct a number of solutions to the appropriate diophantine equation from given ones. The original question was posed by Kummer. There is an intimate connection with the theory of the tetrahedroid and so Kummer surfaces.

Previous work of Vaughan Jones built a bridge between cusp forms and random matrices using the language of von Neumann algebras. Specifically, for a discrete subgroup of $\operatorname{PSL}_2(\mathbb Z)$, there is a natural von Neumann algebra associated to it. Jones showed that for any "tracelike” vector, there is a corresponding (anti)-isomorphism between this algebra and its commutant. Jones also showed that such vectors exist. As he noted, this connection is not useful without an explicit formula, which he posed as an open question. Here, we resolve Jones' question and discuss its context for a broad number theory audience. As we were finalizing this preprint for submission, we learned of an independent paper of Abreu. Both papers employ a closely-related polar normalisation of a kernel orbit to obtain the Jones tracelike vector at the critical parameter. The novelty of the present work lies in its extension to all $1<s\leq13$ and in the detailed analysis of the resulting von Neumann algebra and cusp form correspondences. This paper also arose out of a near solution of the authors from several years ago, which discussions with ChatGPT helped us finalize into a full solution. The relation to Abreu's work, and the exact nature of our use of AI, are detailed in the introduction.