Papers

A classical theorem of Maillet asserts that every nonconstant rational function over $\mathbb{Q}$ maps Liouville numbers to Liouville numbers. In 1984, Mahler asked whether a transcendental entire function can have the same property. We prove a strong smooth counterpart: writing $\mathscr{L}$ for the set of Liouville numbers, there exist orientation-preserving $C^\infty$ diffeomorphisms $f:\mathbb{R}\to\mathbb{R}$, arbitrarily close to the identity and transcendental over $\mathbb{R}(x)$, such that for every real number field $K\subset\mathbb{R}$, every $n\geq 1$, and every $m\geq 0$, \[ D^m(f^{\circ n})(K)\subseteq K, \qquad D^m(f^{\circ n})(\mathscr{L})\subseteq\mathscr{L}. \] In fact, the non-analyticity locus may be prescribed as any nonempty compact perfect nowhere-dense set disjoint from the real algebraic and Liouville numbers. The proof combines Maillet's theorem with an arithmetic refinement of Körner's smooth polynomial sewing method and a rational-germ construction.

We determine two infinite families of poles on the positive real axis for classical single-variable Witten zeta functions. In type $A_r$, for $r \geq 5$, the point $q_r^A = 2(r-4)/(r^2+r-4)$ is a simple pole except in ranks $12$ and $20$, where it is a double pole. In type $D_r$, for $r \geq 4$, the point $q_r^D = (r-3)/(r(r-1)-1)$ is a simple pole except at $D_8$, where it is double. In root-product normalization, we express the simple residues and the leading coefficients of these three double poles in terms of gamma, trigonometric, and Riemann zeta values. For $r \geq 4$, the functions of types $B_r$ and $C_r$ are holomorphic at $q_r^{BC} = (r-3)/(r^2-1)$ except possibly in ranks $7$ and $11$, where any pole is simple. These pole and holomorphy statements arise from quadratic normal Taylor coefficients. For every fixed higher even normal degree, we also determine exactly when the associated continued $B/C$ chamber sum is nonzero; this auxiliary result does not by itself classify poles of the full Witten function. The proofs combine exhaustive support classifications, explicit integration-by-parts identities, and finite chamber relations. No numerical nonvanishing estimate is used.

Let $F$ be a global field and let $p$ be an odd prime with $p\neq\operatorname{char}F$. Assuming first that $\mu_p\subset F$, we record in a uniform global-field form the length-four common-slot chain for equal degree-$p$ symbol classes and emphasize the signed normalization adapted to explicit norm constructions. Thus, from \[(a,b)_p=(c,d)_p\in\textrm{Br}(F)[p]\] one obtains $x,y\in F^\times$ such that \[(a,b)_p=(x^{-1},b)_p=(x,y)_p=(c^{-1},y)_p=(c,d)_p.\] For number fields, the underlying chain is the length-four chain lemma of Gille–Szamuely, based on Tate's simultaneous local–global theorem. The point developed here is that its signed form yields four compatible norm equations that can be used constructively. For the extraspecial central product $H_{p^3}*H_{p^3}$ over a $C_p^4$-Kummer extension, the central-embedding obstruction is $(a,b)_p-(c,d)_p$, while the four norm equations supplied by the chain assemble, under a natural independence hypothesis on the auxiliary Kummer classes, into an explicit factorized radical realization of the central product. Finally, when the ground field does not contain $\mu_p$, we show by a restriction–corestriction argument that the central-embedding obstruction is detected after passage to the cyclotomic extension $F(\mu_p)$. Over that field the problem is Kummer, and the obstruction is again the difference of the two symbol classes.

For a positive integer $a$, let $b(a)$ be the smallest integer $b > a$ such that the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b}$ is smaller than the denominator of $\frac{1}{a} + \frac{1}{a+1} + \cdots + \frac{1}{b-1}$. Recently it was shown that the limit inferior $$\liminf_{a \to \infty} \left(\frac{b(a) - a}{\log a}\right)$$ exists and is positive. Here we find its exact value.

We completely solve Colombo's 1928 determinant problem. For distinct real $x_1,\ldots,x_N$, $N\geq 2$, and an integer $D\geq 1$, we prove that $\det[(x_j-x_i)^D]\neq 0$ if and only if $D\geq N-1$ and either $N$ is even or $D$ is even. The even-exponent case follows from Dyn–Goodman–Micchelli (1986); the remaining odd case is proved by a strict Pfaffian sign theorem. The new odd-exponent theorem and its complete proof chain have also been formalized in Lean 4.

For a finite multiset of integers, repeatedly choose two entries $a,b$ and replace them with $|a^{2}-b^{2}|$. Hickerson and Kleber asked whether, for every integer $n$, the set ${n,n+1,\ldots,n+35}$ can be reduced to zero. We give an explicit reduction. Together with their results for lengths 12 and 24, this gives, for every positive integer $L$, [ {n,n+1,\ldots,n+L-1}\text{ reduces to zero for every }n\in\mathbb Z \Longleftrightarrow 12\mid L\text{ and }L\ge 24. ]

Polymath8b proved that $H_1 = \liminf (p_{n+1}-p_n) \leq 246$. In this paper we show how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain the improved bound $H_1 \leq 240$.

For real $t$ and $x\ge 1$, set \[ S(x,t)=\sum_{n\le x} n^{\ii t}. \] We prove an unconditional inverse theorem relating large values of $S(x,t)$, with $|t|$ large, to zeros of the Riemann zeta function near height $t$. More precisely, if $T\le |t|\le 2T$, $\exp(\sqrt{\log T})\le x\le \sqrt T$, and $|S(x,t)|=x/N$ with $N\le (\log x)^{1/100}$, then for every $cN^6\le L\le (\log x)/2$ a disk centered at $1+\ii\phi$, where $|\phi-t|\ll N$, contains at least $L/360$ zeros of $\zeta(s)$. As a consequence, a quantitative restriction on zeros in a short family of arbitrarily thin fixed windows to the left of the line $\Ree s=1$ yields $S(x,t)\ll x/(\log x)^{1/100}$ in polynomial ranges of $x$. The proof adapts the zero-forcing mechanism of Granville and Soundararajan for large character sums. In the zeta setting the spectral height is shifted by $t$, and an additional residue from the pole of $\zeta(s)$ appears in the Gaussian transform; in the range considered here that residue is exponentially small.

We extend recent work of the author over $\mathbb{Z}$ into the positive characteristic setting of $\mathbb{F}_q[t]$. In particular, for a polynomial $F \in \mathbb{F}_q[t]$ of degree $N$, let $R_k(\alpha)$ denote the exponential sum over $k$-free polynomials $f$ with $\text{deg}(f-F)<K$. For all $s>0$, we prove essentially tight upper and lower bounds for the $s$-th moment of $R_k(\alpha)$ whenever $K > (\frac{1}{2}+\epsilon)N$, and in even shorter intervals when $s>1+\frac{1}{k}$. As an application of these results, we prove a lower bound of order $q^{\frac{K}{6}}$ for the $L^1$-mean of the Möbius-twisted exponential sum over $\mathbb{F}_q[t]$ whenever $K >(\frac{1}{2}+\epsilon)N$.

In this note we count the number of solutions to a non-degenerate quadratic form in eight prime variables of off-diagonal rank $3$. It is a continuation of work by L. Zhao, who counted solutions to forms in at least nine variables, by B. Green, who counted solutions to 'generic' (which implies off-diagonal rank $4$) forms in eight variables, and by J. Dobrowolski, who counted solutions to forms in eight variables of off-diagonal rank at most $2$.

Let the center of a $d$-dimensional Euclidean ball be uniformly distributed modulo $\mathbb Z^d$. We study the resulting number of integer lattice points as the dimension and radius tend to infinity. The critical scale is $4\pi^2R_d^2/(d+2)=\log d$. In a logarithmic window around this scale, the logarithm of the lattice-point count divided by the volume satisfies a central limit theorem. At a fixed offset from the critical scale this yields a genuine lognormal limit. Above the critical scale, the count-to-volume ratio converges to $1$ in measure; its normalized error has a standard normal limit, and its variance satisfies an asymptotic formula.

We study coefficientwise permutation lifts of reduced polynomial representatives of almost perfect nonlinear (APN) permutations from finite fields to Galois rings. The standard permutation criterion shows that a permutation polynomial over $\mathbb F_{2^m}$ admits such a lift if and only if its formal derivative is nowhere zero on the field. In that case, every coefficientwise lift permutes $\operatorname{GR}(2^k,m)$ for every $k>1$. It is natural to conjecture that the reduced representative of an APN permutation admits no such lift. However, Zhang and Haobo supplied a counterexample to our earlier critical-point conjecture, which also refutes this lifting claim. Their degree-$24$ reduced APN permutation over $\mathbb F_{32}$ has a nowhere-zero formal derivative. We record this example and give an exact linear-image and trace-dual criterion for the absence of rational critical points in quadratic functions. We also obtain explicit non-APN bounds from Janwa-Wilson-Rodier surfaces. For every odd degree $d\ge5$ outside the Gold and Kasami-Welch exponent families, results of Hernando-McGuire and Aubry-McGuire-Rodier yield an absolutely irreducible component defined over the ground field. Applying the Cafure-Matera point estimate gives an explicit threshold above which no polynomial of degree $d$ over $\mathbb F_{2^m}$ is APN. This makes the known qualitative nonexistence result effective. An identity between difference tables extends the bound to polynomials $ax+g(x^2)+c$, where $a\ne0$ and $g$ has an odd degree in the stated range. Finally, every cubic permutation polynomial has a rational critical point and therefore admits no coefficientwise permutation lift.

In this paper, we explicitly determine an algebraic Belyi function on a unique smooth projective model $\widetilde{W}$ of the Wiman sextic curve $W$ and describe its complex uniformization in terms of modular functions associated with a certain noncongruence subgroup $\Gamma_{\widetilde{W}} \subset {\rm SL}_2(\mathbb{Z})$. As an application, we give a direct proof of the unbounded denominators conjecture in weight $2$ for $\Gamma_{\widetilde{W}}$. The conjecture is now known in full generality by the work of Calegari, Dimitrov, and Tang, following earlier progress including work of Dong, Lin, and Ng. Our proof, however, uses a degeneration of $\widetilde{W}$ over $\mathbb{F}_{5}$ together with explicit Puiseux series expansions and is substantially different from the methods employed in their work.

The Lagrange spectrum is a classical object in number theory, defined as the set of values of $\liminf_{q\to\infty} q\, \mathrm{dist}(q\alpha,\mathbb{Z})$ where $\alpha$ runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.

For an integer $t\geq-1$, let $\theta_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and set $R_t=\mathbb{Z}[\theta_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(\theta_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We determine $\varepsilon_t$ when $N_t$ is squarefree, $27$, or $343$: the only nontrivial indices in these cases are $\varepsilon_3=3$, $\varepsilon_5=7$, $\varepsilon_{12}=13$, and $\varepsilon_{54}=19$. Local conductor calculations give $\varepsilon_t\mid N_t$ for squarefree $N_t$. For arbitrary $N_t$, a regulator comparison shows that $t\geq2N_t$ implies $\varepsilon_t=1$. When $N_t=p^3$ with $p\neq3$ a rational prime, this bound and the index criterion leave at most four possible parameters with $\varepsilon_t>1$ for each fixed $p$. For arbitrary additive index, we also prove that $13\mid\varepsilon_t$ if and only if $t=12$ or $66$, with unit index $13$ in both cases. The proof determines the rational solutions of a plane quartic equation by an explicit genus-two descent and a local Chabauty argument, proving Louboutin's Conjecture 19 on its integral solutions. When $N_t$ is a rational prime, we determine the Picard kernel, the cardinality and fibers of the ideal class monoid over $\operatorname{Pic}(\mathcal{O}_t)$, and the corresponding integral matrix-conjugacy classes.

For every $1\le d<n$, we prove joint equidistribution, as the height tends to infinity, of $d$-dimensional $K$-subspaces in a rigid adelic space over a number field $K$, together with their archimedean Grassmannian images and the $K$-linear isometry classes of the normalized subspace and quotient. We identify the leading constant, prove no escape of mass from either normalized factor, and obtain global equidistribution against bounded continuous test functions. Over a general number field, the two normalized shapes are coupled by a determinant-class relation; conditional on the determinant class, their limiting law is the product of the Haar-induced probability measures on the corresponding fibers.

We provide a new instance of singular relative Langlands duality, underlying a Rankin-Selberg integral on $\mathrm{GSO}_6$ due to Bump-Friedberg-Ginzburg. We conclude that this integral represents an essentially self-dual object in the relative Langlands program, and we demonstrate that the Langlands dual automorphic integral computes a finite sum of $L$-functions, reflecting stacky structure on the spectral side.

For every quadratic number field $K$, we prove a uniform $3$-unknown Diophantine definition of integer tuples in $\mathcal{O}_K$, allowing finitely many polynomial nonvanishing conditions. This yields an effective $+3$ transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound $D_0\geq1$ such that for every quadratic number field $K$, there is no algorithm that, given \[ P(Y_1,\ldots,Y_{13})\in\mathbb{Z}[Y_1,\ldots,Y_{13}],\quad \mathrm{deg}\ P\leq D_0, \] decides whether $P=0$ has a solution in $\mathcal{O}_K^{13}$. The arithmetic input is a fourth-order Pell–Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas–Wieferich and Wall–Sun–Sun primes. We also establish local surjectivity and $\ell$-adic density for second-order correction terms for norm-one Lucas sequences.

Recently Dixit, Kumar, and Srivastava investigated what they called Rascoe and non-Rascoe partitions. These are defined to be the set of distinct partitions where the length of the partition is a part of the partition and is not a part respectively. In this note we provide a $q$-series theoretic proof of two identities regarding the generating function for unrestricted Rascoe and non-Rascoe partitions fulfilling a request of Dixit, Kumar, and Srivastava. We also prove a conjecture of Beck relating non-Rascoe partitions and the rank of a partition.

Let $q=2^m$, $Q=2^k$, and $1\leq k\leq m-1$. We characterize complete permutation polynomials (CPPs) over $\mathbb{F}_{q^2}$ of the form \[ f(x)=c_0x^{Q+1}+c_1x^{Q+q}+c_2x^{qQ+1} +c_3x^{q(Q+1)},\qquad c_i\in\mathbb{F}_{q^2}. \] We prove that no such CPP exists when $k>1$, and recover the known characterization in the cubic case $k=1$. This completes the classification throughout the stated exponent range. The proof uses the known permutation classification to reduce completeness to linear perturbations of product and monomial models. The required nonpermutation results follow from direct elementary arguments based on the quadratic structure over $\mathbb{F}_2$.

This paper establishes a comprehensive combinatorial and arithmetic framework for mixed Stirling and mixed Bell numbers, bridging partition structures, Touchard polynomials, and prime-power congruences. Furthermore, we develop to $p$-adic valuation theory, proving prime-power Touchard congruences and higher-order modulo-$p^2$ refinements that generalize classical arithmetic properties of combinatorial sequences.

Let $\mathsf{d}$ be the asymptotic density on the positive integers $\mathbb{N}^+$. We provide sufficient conditions on an infinite set $B\subseteq \mathbb{N}^+$ so that for each $\alpha \in [0,1]$ there exists a set $A\subseteq \mathbb{N}^+$ such that $\mathsf{d}(A)=0$ and $\mathsf{d}(A+B)=\alpha$. For instance, if $(b_n: n \in \mathbb{N}^+)$ is the increasing enumeration of $B$, it is sufficient that $\liminf_n b_{n+1}/b_n>1$. Moreover, we show that, if $|B\cap [1,n]|\ll \log(n)^{1-\varepsilon}$ for some $\varepsilon>0$, then for each $\alpha \in [0,1]$ there exists a set $A\subseteq \mathbb{N}^+$ such that $\mathsf{d}(A)=\mathsf{d}(A+B)=\alpha$. Conversely, we show that, for each $\varepsilon>0$, there exists an infinite set $B\subseteq \mathbb{N}^+$ such that $|B\cap [1,n]| \ll n^{2/3+\varepsilon}$ and $\mathsf{d}(A+B) \in \{0,1\}$ for every $A\subseteq \mathbb{N}^+$ such that $A+B$ admits asymptotic density.

Let $A$ be an $n$-element set of positive real numbers, let $FS(A)$ be its set of subset sums, and put $T_n=\binom{n+1}{2}$. For every fixed integer $C\geq-1$ and all sufficiently large $n$, we classify the sets satisfying $$ |FS(A)|\leq T_n+n+C+1. $$ Each such set is commensurable. Its unique primitive integer normalisation $B$ either satisfies $\sum B\leq T_n+n+C$ or belongs to an explicit exceptional family specified by a missing element $m\in\{1,2\}$ and an integer partition of $C+m$ or $C+m+1$. If $P$ denotes the partition function, the exceptional family has exactly $$ P(C+1)+2P(C+2)+P(C+3) $$ primitive dilation classes. We also prove a local inverse theorem for bounded increment excess. If, for sufficiently large $i$, adjoining the largest element to the preceding $i-1$ elements creates only $i+e$ new subset sums, where $e$ is bounded, then the $i$-element set is a dilation of $[1,i+e]_{\mathbb{Z}}$ with exactly $e$ elements deleted. Conversely, every such deletion pattern has increment excess $e$. The proof combines a stabiliser argument in $\mathbb{R}/x\mathbb{Z}$, Kneser's theorem, a quadratic subset-sum bound, and endpoint propagation. These arguments also give effective commensurability and a finite-state encoding. Together with earlier results for $C\leq-2$, this completes the eventual fixed-defect classification for every integer $C$.

Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket quadratics. This comes as a consequence of establishing exponential sum estimates for these sets, utilising a theorem of Green and Tao on the quantitative equidistribution of polynomial orbits on nilmanifolds, closely following the approach of Neale who went on to prove a Waring-type result. We also extend our result to differences avoiding a family of bracket polynomials (also known as generalised polynomials).

We study matrix representations of the Jacobsthal sequence generated by binary 3x3 matrices with determinants 0, 2, and -2, using linear algebraic methods analogous to Fibonacci-type constructions. Explicit formulas for matrix powers are obtained, yielding identities for Jacobsthal numbers, including convolution formulas, trace relations, determinant expressions, and Cassini-type identities. We further derive congruence relations and recurrence formulas, and analyze arithmetic properties such as partial sums and the Sidon-type structure of the sequence. Finally, we prove that exactly three conjugacy classes of binary 3x3 matrices generate the Jacobsthal sequence, providing a unified algebraic framework that connects matrix theory with second-order linear recurrences.

In this article, we introduce a refinement of the method of Heap and Soundararajan to obtain lower bounds for the $2k$-th moment of a broad class of $L$-functions for all real $k\ge 0$. In particular, our method circumvents the need to estimate the twisted moments of $L$-functions.

We introduce $L$-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the $L$-functions arising from unramified coverings of algebraic curves. These $L$-functions are defined using the Riemann–Roch structure on the graph and extend Lorenzini's two-variable zeta function. We show that if two graphs without bridges have isomorphic Jacobians and their $L$-functions agree under the induced correspondence of characters, then their Laplacian lattices coincide. We also show that Lorenzini's zeta function is invariant under contraction of bridges, explaining the necessity of the bridge-free hypothesis in the main theorem. Finally, we give examples showing that neither the Jacobian nor the Lorenzini zeta function alone determine the Laplacian lattice.

We introduce analogues of Diophantine $D(N)$-$m$-tuples in the noncommutative ring $M_2(\mathbb Z)$ of $2\times2$ integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product $$A\circ B=\frac12(AB+BA).$$ Special attention is devoted to upper-triangular integer matrices $UT_2(\mathbb Z)$, where squares admit a particularly simple description. Motivated by the classical connection between representations of $n$ as a difference of two squares and the existence of $D(n)$-quadruples in commutative rings, we investigate the existence of Jordan $D(N)$-quadruples in $UT_2(\mathbb Z)$.

In 1988, Koblitz conjectured an asymptotic formula for the number of primes $p \le x$ for which the reduction of an elliptic curve over $\mathbb{Q}$ has prime order. Building on the work of Balog, Cojocaru, and David, who proved this conjecture on average in 2011, we extend the result to primes $p$ lying in arithmetic progressions. Our asymptotic formula holds uniformly for moduli up to a fixed power of $\log x$. We also show that the resulting average constant matches the theoretical constant predicted by Lee, Mayle, and Wang in 2025 using Galois representations.

Let $p$ be an odd prime, and let $\overline{\mathrm{SL}_2(\mathbb{Z}_p)}$ and $\overline{K_0(p^n)}$ denote the inverse images of $\mathrm{SL}_2(\mathbb{Z}_p)$ and the congruence subgroup $K_0(p^n)$ in the metaplectic double cover $\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)$. For $n\geq2$, we study the subalgebra of the genuine Hecke algebra $H(\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)//\overline{K_0(p^n)},\eta)$ consisting of functions supported in $\overline{\mathrm{SL}_2(\mathbb{Z}_p)}$, where $\eta$ is the genuine extension of either the trivial or the nontrivial quadratic character modulo $p$. We give an explicit basis and a presentation by generators and relations, and prove that this subalgebra is commutative of dimension $2n$. We give the multiplicity-free decomposition of $\operatorname{Ind}_{\overline{K_0(p^n)}}^{\overline{\mathrm{SL}_2(\mathbb{Z}_p)}}\eta$, determining the dimensions of its irreducible constituents and their associated Hecke eigenvalues. This Hecke subalgebra acts on the $(\overline{K_0(p^n)},\eta)$-fixed spaces in irreducible admissible genuine representations of $\widetilde{\mathrm{SL}}_2(\mathbb{Q}_p)$. We compute its action explicitly on newvectors of prescribed $\eta$-type in principal series, Steinberg, even Weil, and supercuspidal representations. For supercuspidal representations, we construct the vectors by compact induction and use Ishimoto's conductor and dimension formulas. Finally, we relate our operator $\mathcal{W}_{n-1}$ to Ishimoto's local realization of Ueda's twisting operator.