Mahler equations for Zeckendorf numeration

Van Lier sequences

On the enumeration of parking functions by leading terms

Given an increasing sequence of positive integers { m n } \left \{ {{m_n}} \right \} , a non-decreasing sequence of positive integers { b n } \left \{ {{b_n}} \right \} , and a measurable, measure-preserving ergodic transformation τ \tau on a probability space ( Ω , F , μ ) \left ( {\Omega ,\mathcal {F},\mu } \right ) , the a.s. convergence of the moving averages T n ( f ) = b n − 1 ∑ k = m n + 1 m n + b n f ( τ k ) {T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )} is considered, for f ∈ L p ( Ω ) f \in {L_p}\left ( \Omega \right ) . A counterexample is constructed in the case of polynomial-like { m n } \left \{ {{m_n}} \right \} .

Diophantine inequalities with mixed powers

Let $R$ be a commutative Noetherian ring and let $M$ be a finitely generated $R$-module. If $I$ is an ideal of $R$ generated by $M$-regular sequence, then we study the vanishing of the first $Tor$ functors. Moreover, for Artinian modules and coregular sequences we examine the vanishing of the first $Ext$ functors.

Let $\varphi $ be Euler’s function and fix an integer $k\ge 0$ . We show that for every initial value $x_1\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+1}=\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for all initial values $x_1,x_2\ge 1$ , the sequence of positive integers $(x_n)_{n\ge 1}$ defined by $x_{n+2}=\varphi (x_{n+1})+\varphi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that k is even.

On A Problem of Erdős and Turán and Some Related Results

Let $A$ be a sequence of positive integers. An element $\alpha$ of $A$ is called an {\it s-near relative prime number} ($s$-near relprime in short) if $\alpha$ is coprime to any distinct element in $A$ except exactly $s$ elements of $A$. In this paper, we study the existence of an arithmetic sequence with no 1-near relprimes.

ON ARCWISE CONNECTEDNESS IN SOLENOIDS

We introduce the concept of coarse-graining invariant sequences (in a commutative ring), and consider their relationship with Mahler equations and automatic sequences.

This introductory account of commutative algebra is aimed at advanced undergraduates and first year graduate students. Assuming only basic abstract algebra, it provides a good foundation in commutative ring theory, from which the reader can proceed to more advanced works in commutative algebra and algebraic geometry. The style throughout is rigorous but concrete, with exercises and examples given within chapters, and hints provided for the more challenging problems used in the subsequent development. After reminders about basic material on commutative rings, ideals and modules are extensively discussed, with applications including to canonical forms for square matrices. The core of the book discusses the fundamental theory of commutative Noetherian rings. Affine algebras over fields, dimension theory and regular local rings are also treated, and for this second edition two further chapters, on regular sequences and Cohen–Macaulay rings, have been added. This book is ideal as a route into commutative algebra.

Let (S,𝔫) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I* be the leading ideal of I in the associated graded ring gr𝔫(S), and set R = S/I and 𝔪 = 𝔫/I. In Goto et al. (2007), we prove that if μ G (I*) = n, then I* contains a homogeneous system {ξ i }1≤i≤n of generators such that deg ξ i + 2 ≤ deg ξ i+1 for 2 ≤ i ≤ n − 1, and ht G (ξ1, ξ2,…, ξ n−1) = 1, and we describe precisely the Hilbert series H(gr𝔪(R), λ) in terms of the degrees c i of the ξ i and the integers d i , where d i is the degree of D i = GCD(ξ1,…, ξ i ). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c 1 + 1, an ascending sequence of positive integers (c 1, c 2,…, c n ), and a descending sequence of integers (d 1 = c 1, d 2,…, d n = 0) such that c i+1 − c i > d i−1 − d i > 0 for each i with 2 ≤ i ≤ n − 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.

On expansions of (Z,+,0)

Approximation by Egyptian fractions and the weak greedy algorithm

We give several applications of the probabilistic method in harmonic analysis and additive number theory. We also give efficient constructions in place of previous probabilistic (existential) proofs. 1. Using the probabilistic method we prove that there exist nonnegative integers $p\sb1,\...,p\sb{N}$ for which$$\left\vert\min\sb{x}\sum\sbsp{j=1}{N}p\sb{j}\cos jx\right\vert = O(s\sp{1/3}),$$as $s\to\infty,$ where $s = \sum\sbsp{j=1}{N}p\sb{j}.$ This improves a result of Odlyzko who proved a similar inequality with the right hand side replaced by $O((s \log s)\sp{1/3}).$ 2. Similarly we prove that there are frequencies $\lambda\sb1 <\cdots<\lambda\sb{N}\in\{1,\...,cN\},$ for c = 2, for which$$\left\vert\min\sb{x}\sum\sbsp{j=1}{N}\cos \lambda\sb{j}x\right\vert = O(N\sp{1/2})$$and that this is impossible for smaller values of the positive constant c. 3. The previous result is used to prove easily a theorem of Erdos and Turan about the density of finite integer sequences with the property that any two elements have a different sum ($B\sb2$ sequences). We also generalize this to $B\sb{2h}$ sequences (of which all sums of 2h elements are distinct). Some dense finite and infinite $B\sb2\lbrack 2\rbrack$ sequences (only two pairs of elements are allowed to have the same sum) are also exhibited. 4. We prove that for any sequence of integers $n\sb1\le\...\le n\sb{N}$ there is a subsequence $n\sb{m\sb1},\...,n\sb{m\sb{r}}$ such that$$\left\vert\min\sb{x}\sum\sbsp{j=1}{r}\cos n\sb{m\sb{j}}x\right\vert\ge C\cdot N,$$where $C>0$ is an absolute constant. Uchiyama had previously proved this with the right hand side replaced by $C\cdot N\sp{1/2}.$ Furthermore, our proof is constructive. We give a polynomial time algorithm for the selection of such a subsequence. 5. set E of positive integers is called a basis if every positive integer can be written in at least one way as a sum of two elements of E. Using the probabilistic method, Erdos has proved the existence of such a basis E for which every positive integer x can be written as a sum of two elements of E, in at least $c\sb1$ log x and at most $c\sb2$ log x ways, where $c\sb1,c\sb2>0$ are absolute constants. We give an algorithm for the construction of such a basis which outputs the elements of E one by one, and which takes polynomial time to decide whether a certain integer is in E or not. 6. We employ the probabilistic method to improve on some recent results of Helm related to a conjecture of Erdos and Turan on the density of additive bases of the integers. We show that for a class of random sequences of positive integers (which satisfy $\vert A \cap \lbrack 1,x\rbrack\vert\ge C\cdot\sqrt{x}),$ with probability 1, all integers in the interval (1,N) can be written in at least $c\sb1$ log x and at most $c\sb2$ log x ways as a difference of elements of $A \cap \lbrack 1, N\sp2\rbrack.$ Furthermore, let $m\sb{k}$ be a sequence of positive integers which satisfies the growth condition$$\sum\sbsp{k=1}{\infty}{\log m\sb{k}\over\sqrt{m\sb{k}}}<\infty.$$We show that, for the same class of random sequences and again almost surely, there is a subsequence $B\subseteq A, \vert B \cap \lbrack 1,x\rbrack\ge C\cdot\sqrt{x},$ such that, for k sufficiently large, each $m\sb{k}$ can be written in exactly one way as a difference of two elements of B.