Tag Archives: arithmetic progression

A nice result on arithmetic progressions

Let \mathbb N_0 denote the set of the non-negative integers. For a subset S\subseteq\mathbb N_0, we can define the ‘counting function’

\pi_S(x):=|\{n\in\mathbb N_0\cap S:n\le x\}|

i.e. \pi_S(x) is the number of elements in S that are at most x. Then we can define the (asymptotic) density of S as

\displaystyle \delta(S):=\lim_{x\to\infty}\frac{\pi_S(x)}{x}

when this limit exists.

For example, the set of the even (resp. odd) numbers has density 1/2. More generally, any arithmetic progression a+d\mathbb N_0:=\{a+dn:n\in\mathbb N_0\} has density 1/d. Another example is the set of prime numbers, which has density 0 by the prime number theorem. From this it also follows that the set of composite numbers has density 1.

Theorem. If a finite union of arithmetic progressions of positive integers has density 1, then it contains all sufficiently large positive integers.

Proof. Let

\displaystyle A=\bigcup_{i=1}^k(a_i+d_i\mathbb N_0)

be a finite union of arithmetic progressions such that \delta(A)=1. Note that (this is the main trick) we can write:

\displaystyle A=\bigcup_{i=1}^k(a_i+d_i\mathbb N_0)=\bigcup_{j=1}^l(b_i+d\mathbb N_0)

for some b_1,\dots,b_l, where d is the least common multiple of d_1,\dots,d_k.

Why can we do this? Consider for example (1+2\mathbb N_0)\cup(2+3\mathbb N_0). We can write this as

\displaystyle \underbrace{(1+6\mathbb N_0)\cup (3+6\mathbb N_0)\cup (5+6\mathbb N_0)}_{1+2\mathbb N_0}\cup \underbrace{(2+6\mathbb N_0)\cup (5+6\mathbb N_0)}_{2+3\mathbb N_0}

\displaystyle =(1+6\mathbb N_0)\cup (2+6\mathbb N_0)\cup (3+6\mathbb N_0)\cup (5+6\mathbb N_0)

And this idea works for any finite union of arithmetic progressions.

It is then clear that \{b_1,\dots,b_l\} has to be a complete set of residues modulo d, because otherwise A would have density strictly less than 1. Therefore A has to contain all positive integers \ge\max\{b_1,\dots,b_l\}. \square

We get the following nice corollary:

Corollary. The set of all composite numbers cannot be expressed as a finite union of arithmetic progressions of positive integers.

Proof. Otherwise by the above theorem all sufficiently large positive integers would be composite, contradicting the fact that there are infinitely many primes. \square

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Finite automata 1: Automatic sets

In one of my recent courses I had to learn about finite automata theory. The main purpose was to understand a generalisation of the celebrated Skolem-Mahler-Lech theorem in positive characteristic. Let \mathbb N_0 denote the set of non-negative integers and K be a field. For a linear recurrence f:\mathbb N_0\to K, define its zero set to be \{n\in\mathbb N_0 : f(n)=0\}. Then the Skolem-Mahler-Lech theorem states the following:

Theorem (Skolem-Mahler-Lech, 1953). The zero set of a linear recurrence f:\mathbb N_0\to\mathbb K, where K is a field of characteristic 0, is a finite union of arithmetic progressions along with a finite set.

Derksen in his 2005 paper was able to provide the correct analogue of this result in positive characteristic for the first time:

Theorem (Derksen, 2007). The zero set of a linear recurrence f:\mathbb N_0\to K, where K is a field of characteristic p>0, is a p-automatic set.

To understand this theorem we need to know about finite automata. In this and the next few posts I want to write about some interesting elementary results about automatic sets. Since my knowledge of computer science is epsilon(!), I will mostly touch on some very basic mathematical aspects.

A deterministic finite-state automaton (DFA) is a directed multigraph \Gamma satisfying certain properties. Firstly, its vertex-set Q (which we call the set of states) contains a special vertex (or state) q_0, called the initial state. We also have a finite non-empty set \Sigma of symbols. Each edge of \Gamma is labelled with a symbol from \Sigma. As a result, we have a transition function \delta:Q\times\Sigma\to Q such that if an edge from q_1 to q_2 is labelled with the symbol a\in\Sigma, then we write \delta(q_1,a)=q_2. Finally, we have a set F\subseteq Q (possibly empty) called the set of accepting or final states.

So \Gamma can be formally written as a quintuple (\Sigma,Q,q_0,\delta,F).

Let \Sigma^* be the free monoid on \Sigma. We can extend the transition function \delta to a map \delta:Q\times\Sigma^*\to Q by setting \delta(q_0,wa)=\delta(\delta(q_0,a),w) for all a\in\Sigma and w\in\Sigma^*. Thus, to \Gamma we can associate a language

\mathcal L_\Gamma:=\{w\in\Sigma^*:\delta(q_0,w)\in F\}\subseteq\Sigma^*.

We call such a language produced from a DFA a regular language. If w\in\mathcal L_\Gamma for some w\in\Sigma^*, we say that w is accepted by \Gamma.

Let \mathbb N denote the set of the positive integers, and let k>1 be an integer. Define \Sigma_k:=\{0,1,\dots,k-1\}. A subset S\subset\mathbb N is said to be k-automatic if there exists a DFA \Gamma=(\Sigma_k,Q,q_0,\delta,F) such that S is precisely the set of those j\in\mathbb N whose base-k representation, when regarded as a string in \Sigma^*, is accepted by \Gamma. If S\subseteq\mathbb N is k-automatic for some k>1, we say S is an automatic set.

For w=w_nw_{n-1}\cdots w_0\in\Sigma_k^* with each w_i\in\Sigma_k we define

[w]_k:=w_nk^n+w_{n-1}k^{n-1}+\cdots+w_0.

In this case we also define |w|:=n+1 to be the length of w.

So S\subseteq\mathbb N is k-automatic precisely when S=\{[w]_k:w\in\mathcal L_\Gamma\} for some DFA \Gamma.

Note that we can set \delta(q,0) to be in Q\backslash F for every q\in Q (if F=Q, just create a new state). Thus we may without loss of generality assume that all our DFAs only accept words without leading zeros.

A basic result about regular languages is the pumping lemma:

Lemma (Pumping lemma). If \mathcal L is an infinite regular language, then there exist words a,b,c with |b|\ge 1 such that ab^nc\in\mathcal L for all n\ge 0.

Proof. Since \mathcal L is infinite, there are words of arbitrarily large length in \mathcal L. Let w=w_nw_{n-1}\cdots w_0\in\mathcal L be a word of sufficient length. Then there exist 0\le i<j\le n such that w_i\cdots w_0=w_j\cdots w_0\in\mathcal L. Taking a=w_n\cdots w_j, b=w_{j-1}\cdots w_i, c=w_{i-1}\cdots w_0 shows that ab^nc\in\mathcal L for all n\ge 0. Moreover, since i<j, it follows that |b|\ge 1. \square

In the following posts we are going to use the pumping lemma, together with other machinery, to derive some interesting results on the automaticity of certain sets.

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Filed under Algebra, Combinatorics, Number theory