Additive number theory and the ring of quantum integers

In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalises strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some ...

For every positive integer h, the representation function of order h associated to a subset A of the integers or, more generally, of any group or semigroup X, counts the number of ways an element of X can be written as the sum (or product, if X is nonabelian) of h not necessarily distinct elements of X. The direct problem for representation functions in additive number theory begins with a subset A of X and seeks to understand its representation functions. The inverse problem...

These are notes to accompany four lectures that I gave at the School on Additive Combinatorics, held in Montreal, Quebec between March 30th and April 5th 2006. My aim is to introduce ``quadratic fourier analysis'' in so far as we understand it at the present time. Specifically, we will describe ``quadratic objects'' of various types and their relation to additive structures, particularly four-term arithmetic progressions. I will focus on qualitative results, referring the...

In this paper, we extend the lattice Constructions $D$, $D'$ and $\overline{D}$ $($this latter is also known as Forney's code formula$)$ from codes over $\mathbb{F}_p$ to linear codes over $\mathbb{Z}_q$, where $q \in \mathbb{N}$. We define an operation in $\mathbb{Z}_q^n$ called zero-one addition, which coincides with the Schur product when restricted to $\mathbb{Z}_2^n$ and show that the extended Construction $\overline{D}$ produces a lattice if and only if the nested codes...

We show that ${\mathbb Z}$ is definable in ${\mathbb Q}$ by a universal first-order formula in the language of rings. We also present an $\forall\exists$-formula for ${\mathbb Z}$ in ${\mathbb Q}$ with just one universal quantifier. We exhibit new diophantine subsets of ${\mathbb Q}$ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof of the fact that the set of non-squares is diophantine. Finally, we show that there i...

This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and affine quantum groups at zero level $U_q(\hat{\mathfrak{g}})_{c=0}$ corresponding to an arbitrary finite-dimensional semisimple Lie algebra $\mathfrak{g}$. At the intermediate step we construct the embedding of the quantum groups into the a...

It is known that for an IP^{*} set A in (\mathbb{N},+) and a sequence \left\langle x_{n}\right\rangle _{n=1}^{\infty} in \mathbb{N}, there exists a sum subsystem \left\langle y_{n}\right\rangle _{n=1}^{\infty} of \left\langle x_{n}\right\rangle _{n=1}^{\infty} such that FS\left(\left\langle y_{n}\right\rangle _{n=1}^{\infty}\right)\cup FP\left(\left\langle y_{n}\right\rangle _{n=1}^{\infty}\right)\subseteq A. Similar types of results have also been proved for central^{*} sets...

Science and mathematics help people better to understand world, eliminating different fallacies and misconceptions. One of such misconception is related to arithmetic, which is so important both for science and everyday life. People think that their counting is governed by the rules of the conventional arithmetic and that other kinds of arithmetic do not exist and cannot exist. It is demonstrated in this paper that this popular image of the situation with integer numbers is i...

Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose prope...

Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.