Related papers: On arithmetic progressions in symmetric sets in fi…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$…

We show that for every $\varepsilon>0$ there is an absolute constant $c(\varepsilon)>0$ such that the following is true. The union of any $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences must consist of at…

Let K be an algebraically closed field of prime characteristic p, let N be a positive integer, let f be a self-map on the algebraic torus T=G_m^N defined over K, let V be a curve in T defined over K, and let x be a K-point of T. We show…

We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…

Let $(u(n))_{n\in\mathbb{N}}$ be an arithmetic progression of natural integers in base $b\in\mathbb{N}\setminus \{0,1\}$. We consider the following sequences: $s(n)=\overline{u(0)u(1)\cdots u(n) }^b$ formed by concatenating the first $n+1$…

Quantum information processing in the presence of continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. SU(d) is a continuous group of particular interest since it represents a fundamental…

We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) = 1$.

We study approximation of embeddings between finite dimensional L_p spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

We consider, over both the integers and finite fields, Szemer\'{e}di's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of…

We prove that the the discrepancy of arithmetic progressions in the $d$-dimensional grid $\{1, \dots, N\}^d$ is within a constant factor depending only on $d$ of $N^{\frac{d}{2d+2}}$. This extends the case $d=1$, which is a celebrated…

For a prime $p\ge 5$ let $q_0,q_1,\ldots,q_{(p-3)/2}$ be the quadratic residues modulo $p$ in increasing order. We study two $(p-3)/2$-periodic binary sequences $(d_n)$ and $(t_n)$ defined by $d_n=q_n+q_{n+1}\bmod 2$ and $t_n=1$ if…

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic…

We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of…

Finite-density QCD and many other field theories with sign problems have a $\mathcal{PT}$-type symmetry. After a brief introduction to $\mathcal{PT}$-symmetric field theories, a real dual representation for $\mathcal{PT}$-symmetric scalar…

We study some sum-product problems over matrix rings. Firstly, for $A, B, C\subseteq M_n(\mathbb{F}_q)$, we have $$ |A+BC|\gtrsim q^{n^2}, $$ whenever $|A||B||C|\gtrsim q^{3n^2-\frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(\mathbb{F}_q)$…

If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 +…

A $(v, k, \lambda)$ symmetric design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block. Symmetric designs fulfilling this property have the…

Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain…