Papers

For $n\geq1$, choose one residue class $r_k\pmod{k}$ for every $2\leq k\leq n$, and retain those $m\in\{2,\ldots,n+1\}$ for which $m\not\equiv r_k\pmod{k}$ whenever $k<m$. Let $N(n)$ be the number of distinct survivor sets obtained in this way. We prove \[ \log 2\leq \liminf_{n\to\infty}\frac{\log N(n)}{n/\log n} \leq \limsup_{n\to\infty}\frac{\log N(n)}{n/\log n} \leq2\log 2. \] If $A_{\rm adm}(n)$ denotes the number of subsets of $\{2,\ldots,n+1\}$ that omit at least one residue class modulo every prime, then our main comparison is \[ \log N(n)=\log A_{\rm adm}(n)+o\!\left(\frac{n}{\log n}\right). \] We also obtain an asymptotic formula for the number of distinct intersections of survivor sets with the primes, and an exact recurrence $N(n+1)=N(n)+E(n)$, where $E(n)$ counts the survivor sets that can be extended by the new point $n+2$.

For an elliptic curve $E/\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\mathbb{Q})[p^\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\mathbb{Q})[p^\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.

For integers $t\geq -1$, let $\theta_t$ be the largest real root of the Shanks polynomial $g_t(X)=X^3-tX^2-(t+3)X-1$ and put $K_t=\mathbb{Q}(\theta_t)$. We classify the points on $E_n:y^2=x^3-n^2x$ over $K_t$ with abscissa $\theta_t-1$ when $n$ is a positive integer and $E_n(\mathbb{Q})$ has rank zero. We determine the four pairs $(d,t)$ with $d>0$ rational for which $(\theta_t-1)/d^2$ is an abscissa on $E_3(K_t)$. We also exclude the abscissa $\theta_t-1$ on $E_5(K_t)$ and $E_6(K_t)$ and prove that only finitely many parameters $t$ admit this abscissa for each fixed positive integer $n$.

Recently, Kang, Kim, and Lee \cite{Kang2026} developed a unified moment-trace framework for symmetric partition statistics using complete Bell polynomials and their inversion formula. In this paper, we apply this framework to crank statistics for $t$-core partitions and overpartitions. For $t\in\{5,7,11,17,19\}$, we show that the normalized even crank moment generating functions for $t$-core partitions admit partition-trace representations in terms of the functions $D^{(t)}_{2s}(\tau)$, together with suitable Bernoulli-number shifts. We also establish inverse trace formulas that recover $D^{(t)}_{2s}(\tau)$ from the corresponding normalized even crank moments. For overpartitions, we obtain analogous trace and inverse-trace identities for the normalized even moments associated with the first and second residual crank generating functions. As applications, we use complete Bell polynomials and their inversion formula to obtain explicit expressions for the $t$-core partition numbers and overpartitions number in terms of sums involving divisor function.

We investigate the properties of matrices obtained from a geometric transformation of the first $N$ rows of Pascal's triangle. For $N > 2$, their congruence properties form a Pascal tiling, that is, a perfect alternation between entries congruent to $0 \pmod{N}$ and the others, if and only if $N$ is prime. This result yields an alternative proof of the classical congruence $L_N-1\equiv 0 \pmod{N}$ for prime $N$, where $L_N$ denotes the $N$th Lucas number. Within the framework of the Pascal tiling theorem, this congruence can be expressed as a sum of entries lying along a diagonal of one of the matrices under consideration; when $N$ is prime, each of these entries is congruent to $0 \pmod{N}$. By contrast, for Fibonacci pseudoprimes, the sum remains congruent to $0 \pmod{N}$ while at least one of its terms is not. Finally, these results are interpreted in terms of decompositions of binomial coefficients and extended to multinomial coefficients, leading to a study of the associated symmetries. This perspective highlights the case where $N$ is a prime power and clarifies the conditions under which a Pascal tiling arises.

We study surjectivity of the geometric Sen map for perfectoid $G$-torsors over smooth rigid analytic spaces over $\mathbb{C}_p$, where $G$ is a compact $p$-adic Lie group. We construct an affinoid perfectoid $\mathbb{Z}_p^2$-torsor over a smooth curve whose Sen map is not surjective, thereby disproving Rodríguez Camargo's conjectured implication from perfectoidness to surjectivity. We prove that the Sen map of every perfectoid $G$-torsor over a smooth curve is nevertheless surjective on a Zariski open dense subset. Finally, we show that locally analytic decompletion implies Sen surjectivity for diamondian affinoid perfectoid torsors.

We prove Todd's relation conjecture: all $\mathbb{F}_q(\theta)$-linear relations of Thakur's multiple zeta values are generated from the fundamental binary relation by the operators $\mathcal{B}^{\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\mathrm{S}}\circ \mathcal{C}^{\mathrm{S}}$; moreover they are also generated by $\mathcal{B}^{\ast\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\ast\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$. The $\mathcal{B}^{\ast\mathrm{S}}$-part of this conjecture has been proved by Chang, Chen and Mishiba. We prove the whole conjecture for Carlitz multiple polylogarithm values, which implies the $\mathcal{B}^{\mathrm{S}}$-part for multiple zeta values. We also determine all fixed relations and binary relations. Let $\mathfrak{BR}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(\theta)$-linear space spanned by binary relations of weight $w$, and let $\operatorname{Fix}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(\theta)$-linear space spanned by fixed relations. We derive generating functions $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\operatorname{Fix}^{\mathrm{S}}_w\bigr)x^w=\frac{x^{q+1}(1-x)}{(1-2x)(1-2x+x^{q+1})}$ and $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\mathfrak{BR}^{\mathrm{S}}_w\bigr)x^w=\frac{x^q(1-x)}{(1-2x)(1-2x+x^{q+1})}.$ Our results are based on the recent work of Im-Kim-Ngo Dac on the $\mathbb{F}_q$-linear relations of Thakur's multiple zeta values, and a system of transfer theorems between multiple zeta values and multiple polylogarithm values.

We show that replication carries congruence symmetry down an explicit level tower. If a normalized replicable function $f$, holomorphic on the upper half-plane, is invariant under $\Gamma_0(N)$, then its $n$th replicate is invariant under $\Gamma_0(N/(N,n))$. Thus every replicate whose index is divisible by $N$ is the normalized modular invariant $J=j-744$. This gives a direct and classification-free proof of $J$-finality for congruence-invariant replicable functions. The argument uses only the replication identities and an elementary generation theorem for congruence subgroups; it requires neither complete replicability nor arithmetic hypotheses on the Fourier coefficients. We then apply the descent law to completely replicable functions of finite replication order. Their replication towers have a canonical terminal replicate, and the only possible terminal functions are $J$, $q^{-1}$, and $q^{-1}+q$. Moreover, a finite-order completely replicable function is $J$-final precisely when its terminal replicate is $J$, and any symmetry beyond translations forces this alternative.

Let $K/\mathbb Q$ be an imaginary quadratic extension and $p$ an odd prime. Write $\rho=\operatorname{Ind}_{G_K}^{G_{\mathbb Q}}\chi$, where $E/\mathbb Q_p$ is a finite extension and $\chi:G_K\to\mathcal O_E^\times$ is a continuous character. For a fixed $r\in\mathbb Z\setminus\{0\}$, let $\pi_{\rho,r}(X)$ denote the number of rational primes $\ell\le X$ such that $\rho$ is unramified at $\ell$ and $\operatorname{tr}\rho(\operatorname{Frob}_\ell)=r$. Let $a,b$ be the two weights of $\chi$ at $p$. We prove that if $(a,b)\notin\mathbb Q^2$, then $\pi_{\rho,r}(X)\ll_{\rho,r,\varepsilon}X^\varepsilon$ for every $\varepsilon>0$, while if $(a,b)\in\mathbb Q^2\setminus\mathbb Z^2$, then only finitely many such primes occur. These bounds are substantially sparser than the classical CM Lang–Trotter scale. The main input in the non-rational case is a rigidity theorem for algebraic curves in the \(p\)-adic analytic trace locus, combined with rigid-analytic Pila–Wilkie counting; the rational non-integral case is treated by a local ramification argument.

Let $X$ be an affinoid rigid analytic space over a $p$-adic field, equipped with a suitable étale map to a unit polydisk. We provide a formalism of imperfect relative period rings over $X$, which lie inside the corresponding perfect relative period rings constructed by Kedlaya–Liu. Then we establish a relative version of the Fontaine–Cherbonnier–Colmez equivalence between $p$-adic local systems on $X$ and étale $(\varphi, \Gamma)$-modules over such imperfect relative period rings, generalizing previous works of Andreatta–Brinon and others. Using this equivalence, we construct $p$-adic differential equations attached to de Rham local systems, carrying both geometric and arithmetic differential operators. This generalizes the work of Berger to the relative geometric setting. Along the way, we study a relative Fontaine–Sen theory on the decompletion of $\Gamma$-modules over the relative $\mathbf{B}_{\mathrm{dR}}^+$-period rings.

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence. We show that a multiplicative function $f$ satisfying \[ f(F_n+F_m)=f(F_n)+f(F_m)\qquad(n,m\geq 1) \] need not be the identity function, even when $f$ takes positive integer values. This answers negatively a question posed by Spiro in 1992. The smallest example presented here is obtained from $F_{31}=557\cdot 2417$. We prove the required divisibility equivalence, formulate an abstract prime-signature construction, and give a practical criterion producing further examples. We also describe the AI-assisted search that led to the construction and provide a reproducible certificate checker.

We prove some new identities for homogeneous symmetric polynomials when evaluated at roots of unity. Some of these formulas are applied to obtain new insights onto cyclotomic polynomials.

Motivated by a question of Corvaja and Zannier, we investigate whether the projections on the real torus of three real algebraic plane curves admit infinitely many triple intersections. We analyze the case of three lines in detail, obtaining a complete classification. For higher-degree curves, we derive several finiteness results using Siegel's and Levin's theorems on integral points on curves. On the other hand, we construct explicit examples of higher-degree curves with infinitely many triple intersections.

Let $G$ be a finite abelian group of order $N$, $S \subseteq G \setminus \{0\}$ a symmetric connection set, and $K$ a field with $char(K) \nmid N$. The spectral algebra $\mathscr{A_K}(Cay(G,S)) = K[A]$ generated by the adjacency matrix of the Cayley graph is proved to decompose, via the character-orbit decomposition, as a semisimple product of field extensions of $K$, one factor for each $Gal(\overline{K}/K)$-orbit of the eigenvalues $\lambda_\chi = \sum_{s \in S} \chi(s)$. The proof uses the abelian discrete Fourier transform to diagonalise $A$, the Galois action on the character group $\widehat{G}$ to partition eigenvalues into orbits, and the Chinese Remainder Theorem to convert the squarefree minimal polynomial into a Wedderburn product. The dimension of $\mathscr{A_K}(Cay(G,S))$ equals the number of distinct eigenvalues, the idempotent count is $2^r$ where $r$ is the orbit number, and primitive idempotents are computed explicitly via the Bezout algorithm in $K[x]$. Over $\mathbb Q$, every Wedderburn summand is a real subfield of the cyclotomic field $\mathbb Q(\zeta_N)$. New results include: a tensor-product comparison for Cartesian products of Cayley graphs; a systematic analysis of the spectral algebra for elementary abelian groups $(\mathbb Z/p)^k$ (rational for $p \le 3$, requiring real cyclotomic extensions for $p \ge 5$); and a worked orbit analysis for non-cyclic groups including $\mathbb Z/6 \times \mathbb Z/2$ and $\mathbb Z/5 \times \mathbb Z/2$. The cyclic case recovers the companion result $\mathscr{A}_{\mathbb{Q}}(C_n) \cong \prod_{d \mid n} \mathbb Q(\zeta_d)^+$; the Hamming cube gives $\mathscr{A}_\mathbb{Q}(\mathbb Q_k) \cong \mathbb{Q}^{k+1}$. The characteristic-$p$ case is also treated.

Fix $r\ge2$ and let $D_k$ be the natural density of the $n$ such that none of $n+1,\dots,n+k$ is $r$-free. We prove that $\log(1/D_k)=\frac{r-1}{\zeta(r)}k\log k+\frac{r}{\zeta(r)}k\log\log k-C_rk+o(k)$, where $C_r=\frac{1}{\zeta(r)}((r-1)\log\zeta(r)+r-1+r\log\frac{\pi}{r\sin(\pi/r)})$; for $r=2$ this is $C_2=\frac{6}{\pi^2}(1+\log\frac{\pi^4}{24})=1.4595513\dots$. This determines Mirsky's constant $\alpha(h)$, the density of the $n$ with $s_{n+1}-s_n=h$ in the squarefree numbers, to the same precision. Previously the only known estimate was $\log\alpha(h)\le-\frac54 h\log\log h+O(h)$; ours improves this by a factor of order $\log h/\log\log h$, and is two-sided. The upper bound rests on the observation that the primes $p\le z$ with $\prod_{p\le z}p^r\le\sqrt k$ sieve a window of length $k$ with no freedom of alignment, so that at least $(1/\zeta(r)+o(1))k$ positions must be covered by $r$-th powers of distinct primes exceeding $k^{1/r}$. The resulting sum over injections is $m!e_m$ with $m\sim k/\zeta(r)$, evaluated by a saddle point. For the lower bound the Chung-Erdos inequality recovers the full contribution of $m!e_m$ rather than merely its largest term, which is what makes the linear coefficient explicit. As a consequence the first-moment threshold for gaps between $r$-free numbers is $\frac{\zeta(r)}{r-1}\frac{\log x}{\log\log x}(1-\frac{(1+o(1))\log\log\log x}{(r-1)\log\log x})$; for $r=2$ the constant is $\pi^2/6$, that of Erdos's 1951 lower bound. We also prove that every covering system of length $k$ has modulus at least $\exp((\frac{r}{\zeta(r)}+o(1))k\log k)$, so the Chinese remainder construction yields only $\zeta(r)/r$, half of $\pi^2/6$ when $r=2$. Erdos's bound is thus equivalent to the assertion that the first run of $k$ consecutive non-$r$-free integers occurs at the first-moment threshold, not at the scale set by the minimal modulus of a covering system.

Recently, on the non-trivial zeros of the Riemann zeta function, it is obtained by Alpöge and Furman that more than 67.25% of the zeros are simple and on the critical line, and more than 83.62% are distinct. Later, Lamzouri gave a different and more direct proof. In this article, we will use the method of Lamzouri to give lower bounds on the number of the non-trivial zeros of the Riemann zeta function in short intervals. To prove the main result, we establish Montgomery's theorem on the pair correlation of zeros of the zeta function in short intervals by following the approach of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, and then use Lamzouri's inequality on any finite multiset of complex numbers which is invariant under complex conjugation.

Majorization is a fundamental tool for comparing how "spread out" the entries of two vectors are. Key majorization results were obtained for the integers by Hardy, Littlewood and Pólya and for $\mathbb{Z}_p$ by Lev. In this paper, we establish a powerful majorization theorem in $\mathbb{Z}_2^n$ that is an analogue of Lev's result in $\mathbb{Z}_p$. Our proof is based on a novel use of a compression argument that optimizes certain sums of additive representation counts. Our majorization theorem has several applications in additive combinatorics. We resolve the question: given subsets $A_1,\ldots,A_k\subseteq\mathbb{Z}_2^n$ of prescribed sizes, when is the function $r_k(A_1,\ldots,A_k)$, which counts the number of additive $k$-tuples in $A_1\times\cdots\times A_k$, maximized? We establish the corresponding minimization result and a characterization of all extremizers for $r_k(A,\ldots,A)$ when $k$ is odd. When $k=3$, this quantity is the number of Schur triples in $A$; as a special case, we recover a theorem of Samotij and Sudakov on Schur triples. We also obtain a convexity inequality and use it to prove an analogue of Pollard's theorem that strengthens and extends a well-known result of Bollobás and Leader in the $\mathbb{Z}_2^n$ setting.

Kanade and Russell conjectured seventeen Rogers–Ramanujan type identities of modulus 12. Eleven of these identities involving triple sums were proved by Bringmann–Jennings-Shaffer–Mahlburg and by Rosengren. Motivated by these works and using similar methods, we settle all the six remaining conjectures including two triple sum identities labeled $I_{5a}$ and $I_{6a}$ originated from Russell's thesis and four quadruple sum identities labeled 7, 7a, 8 and 8a. Our proof of the triple sum identities combines linear recurrences, $q$-difference equations, and $q$-series summation formulas. For the quadruple sum identities, we represent the sums as contour integrals whose integrands are infinite products and evaluate them by residue calculus. The resulting residues reduce to single sum cubic basic hypergeometric series, and we are able to express them as infinite products.

Let $p$ be an odd prime, and let $a_n$ be the reduction modulo $p$ of the $n$th coefficient of the Artin-Hasse exponential. We study the weighted $p$-section convolutions $W_k$ formed from the coefficients $a_{kp}$. We prove the conjecture of Avitabile and Mattarei for $1<k<p$ and extend it to a single global power-series identity determining the entire sequence $(W_k)_{k\geq 0}$. This identity yields an explicit base-$p$ digit formula; in particular, $(W_k)$ is $p$-automatic and admits an explicit finite-state evaluator. Independently, the exact conjugated $p$-section equation gives a strict $p$-adic fixed-point iteration for the logarithmic derivative of the $p$-section. Big-Witt reconstruction then recovers the $p$-section, and hence all coefficients of the Artin-Hasse exponential, to arbitrary prescribed $p$-adic precision by a purely modular procedure.

We develop a theory of logarithmic $A_{\mathrm{inf}}$-cohomology with coefficients for a class of $p$-adic log formal schemes that are “sufficiently log smooth”, where the coefficients are given by relative log BKF modules. Then we establish comparison isomorphisms with étale, de Rham, and crystalline cohomology, and also extend these results to the derived setting. As an application, we give a new proof of the $C_{\mathrm{st}}$ conjecture for semistable local systems. The proof also uses the prismatic interpretation of semistable local systems established by Du–Liu–Moon–Shimizu.

We study the equidistribution of small points over finitely generated fields, in connection with a conjecture of Yuan–Zhang. We first give a counterexample showing that numerical smallness, defined using Moriwaki heights for all polarizations, does not imply equidistribution at every valuation. We then prove that numerically small points do equidistribute at every fully transcendental valuation. In particular, when \(F\) is the function field of a curve \(B/\mathbb{Q}\), these valuations correspond to points of types \(2\), \(3\), and \(4\) in the Berkovich analytification of \(B_{\mathbb{Q}_p}\).

In this note, we give a counterexample to the assertion of Problem 10.4 in Bugeaud's monograph Distribution modulo one and Diophantine approximation, which goes back to Mendès France. The problem states that the spectrum of the sequence $(\xi\alpha^n)_{n\ge1}$, that is, the set of irrational $\theta\in(0,1)$ for which $(\xi\alpha^n-n\theta)_{n\ge1}$ is not uniformly distributed modulo one, is at most countable for all real $\xi\ne0$ and $\alpha>1$. More precisely, we prove that for every real $\alpha>1$ there are $2^{\aleph_0}$ real numbers $\xi>0$ for which the spectrum of $(\xi\alpha^n)_{n\ge1}$ contains one and the same uncountable set.

We exhibit an explicit pair $(A,B)$ of integral symmetric $9\times9$ matrices, defining a nonsingular pair of quadratic forms over $\mathbb{Q}$, such that no member of the rational pencil $\lambda q_A+\mu q_B$ has Witt index $4$ over $\mathbb{Q}$. This refutes a conjecture from \cite{Que16a}, which predicted that every nonsingular pair in $n$ variables generates a pencil containing a form of Witt index $\lceil (n-1)/2\rceil$. The obstruction is purely $2$-adic and affects the whole pencil at once: every member has Witt index exactly $3$ over $\mathbb{Q}_2$. The proof is finite and elementary: a parity argument on $\det(\lambda A+\mu B)$, a congruence lemma reducing $\mathbb{P}^1(\mathbb{Q}_2)$ to the twelve classes of $\mathbb{P}^1(\mathbb{Z}/8)$, and a verification at each class, in which the anisotropy verdict is certified in two independent ways. All scripts are provided in the GitHub repository.

Let \(N\ge2\) and let \(p\) be its least prime divisor. We prove that every nonzero polynomial with nonnegative real coefficients divisible by \(\Phi_N\) has support diameter at least \((p-1)N/p\). Equality holds precisely for positive scalar multiples of monomial shifts of the \(p\)-term geometric sum \(\sum_{j=0}^{p-1} X^{jN/p}\), thereby proving a conjecture of Steinberger. The proof turns cyclotomic divisibility into the vanishing of the first \(p-1\) Fourier moments of a positive measure on the circle and then applies a classical extremal trigonometric polynomial. As a consequence, we establish the Coven–Meyerowitz diameter bound under their tiling conditions and determine its equality cases. Longer initial intervals of vanishing Fourier coefficients yield stronger diameter bounds, including an explicit refinement in terms of the prime-power divisor sets. The extremal trigonometric polynomial also yields a quantitative concentration estimate for measures and cyclotomic multiples with near-minimal support diameter.

We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice $\Gamma$ for which $$H^1_{\mathrm{cusp}}(\Gamma\backslash\mathbb B^2,\mathbb C)\ne 0.$$ In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for $\mathrm{U}(3)$. A cohomological criterion proved previously by the author then gives the residual-finiteness result. Residual finiteness also yields multiplier systems of arbitrary denominator on suitable finite-index subgroups.

In this paper we study a naturally occurring variant of Ramanujan polynomials with a Bernoulli convolution structure, $S_v$, related to the double Mordell-Tornheim series. In particular, we prove for even $v$ all zeros of $S_v$ lie on the complex unit circle, and for even $v \ge 4$ they are simple with the exception of roots at $1, -1$, which are double roots.

We present solutions to six problems concerning sets of nice recurrence and van der Corput (vdC) sets. We answer a question of Moreira, repeated by Fish and Skinner, by showing that every set of ergodic nice recurrence is a set of nice recurrence. We answer questions of Bergelson as well as Bergelson and Lesigne by showing that every set of nice recurrence is a nice vdC-set, that the family of nice vdC-sets is partition regular, and that the family of sets of nice recurrence is partition regular. We mention that Chapman independently proved that sets of nice recurrence are partition regular. We answer a question of Kelly and Lê, a special case of which was previously asked by Peres, and show that for every nontrivial compact metrizable group $K$, every $K$-vdC set is also a vdC set. Lastly, we answer a question of Farhangi, Rodríguez, and Tucker-Drob and show that for any countably infinite discrete group $G$, a vdC set in $G$ is a set of operatorial recurrence in $G$. The initial solutions to these problems were produced by an AI system. The authors checked their correctness, simplified and reformulated them as clear, human-readable proofs, and situated the resulting contributions within the existing literature.

In this paper, we revisit the work of Berkovich and Paule on variants of the Andrews-Gordon identities. We find a generalization of their principal polynomial identity and, as a consequence, obtain new $\vec M$ versions of the Andrews-Gordon identities. More precisely, for non-negative integers $M_1\ge M_2\ge M_3\ge\ldots\ge M_\nu$, we show that a broad class of multi-sums of the form $$ \sum\limits_{\mathbf n} \frac{ q^{ N_1^2+\cdots+N_\nu^2 - M_1N_1 - M_2N_2 - \cdots - M_\nu N_\nu } }{ (q)_{n_1}(q)_{n_2}\cdots(q)_{n_\nu} } $$ can be expressed as a sum of products. Above, we use standard notations for $q$-Pochhammer symbols and $N_i = \sum\limits_{k=i}^{\nu}n_k$ for $1\le i\le \nu$.

A family $\mathcal F$ of finite subsets of $\mathbb Z$ is packed into $[N]$ if suitable integer translates of its members are pairwise disjoint subsets of $[N]$. We study two prescribed-difference packing problems of Alon, Dębski, Grytczuk and Przybyło for the arithmetic progressions $A_d=\{d,2d,\ldots,\lfloor n/d\rfloor d\}$ and $B_d=\{d,2d,\ldots,nd\}$. Let $m(n)$ and $M(n)$ denote the corresponding minimum packing lengths, with subscripts indicating restrictions on the admissible differences, and let $\mathcal P(x)$ denote the set of primes at most $x$. The key ingredient is a residue-class selection scheme that encodes pairwise intersection constraints by cyclic intervals. A lattice-covering argument yields a simultaneous admissible choice, permitting substantial overlap of the containing intervals and providing the sharp upper bounds needed for the bounded-diameter family and the prime-difference equal-cardinality family. For the bounded-diameter family, we prove $m(n)\sim m_{\mathcal P(\sqrt n)}(n)\sim 4n^{3/2}/(3\log n)$. For the equal-cardinality family with prime differences, we prove $M_{\mathcal P(n)}(n)\sim n^3/(6\log n)$. For the full equal-cardinality family, we show $M(n)\ge \left(\frac{19}{108}-o(1)\right)n^3/\log n$. Together with corresponding estimates for restricted ranges of differences, these results prove several conjectures of Alon–Dębski–Grytczuk–Przybyło and disprove others.

In this article, we prove the dynamical Mordell-Lang conjecture for product of endomorphisms on curves defined over complex number field. Moreover, we proved that there is some effectively computable on the the common difference of the arithmetic progressions.