Expansion, Divisibility and Parity: An Explanation

Abstract

After seeing how questions on the finer distribution of prime factorization—considered inaccessible until recently—reduce to bounding the norm of an operator defined on a graph describing factorization, we will show how to bound that norm. In essence, the graph is a strong local expander, with all eigenvalues bounded by constant factor times the theoretical minimum (i.e., the eigenvalue bound corresponding to Ramanujan graphs). The proof will take us on a walk from graph theory to linear algebra and the geometry of numbers, and back to graph theory, aided, along the way, by a generalized sieve. This is an expository paper; the full proof has appeared as a joint preprint with M. Radziwił.

Appendix 1: Sieves

What we must address now may be seen as a technical task. However, the way we will address it most likely has more general applicability.

Our task in this appendix is to show how to exclude from our set $X\subset \textbf{N}$ all integers n that could give rise to premature revenants, that is, edge lengths $p_i=p_i'$ with $i'-i$ small such that $p_j\ne p_i$ for some $i<j<i'$. (Without this last condition, we would be counting not only “revenants” but also mere repetitions.) As we already commented, it is enough to exclude the set $Y_\ell$ of all integers n such that there exist $p\in \textbf{P}$, $p_1,\dotsc ,p_l\in \textbf{P}$ with $p_i\ne p$ and $\sigma \in \{-1,1\}^l$, $l\le \ell$, for which

$$\begin{aligned} \begin{aligned} p|n, p_{1}&|n, p_{2} | n + \sigma _1 p_1, \dotsc , \\p_l&|n+\sigma _1 p_1 + \dotsc + \sigma _{l-1} p_{l-1}, p|n+\sigma _1 p_1 + \dotsc \sigma _l p_l.\end{aligned}\end{aligned}$$

(9)

It is actually easy (and in fact an exercise for the reader) to show that $Y_{\ell }$ is quite small—not much larger than $O(\mathscr {L})^{\ell } N/H_0$. (Outline: the probability that a given divisor p of $\sigma _1 p_1 + \dotsc + \sigma _l p_l$ divide a random n is about 1/p, which is at most $1/H_0$; the probability that there be $p_1,\dotsc ,p_l$ as above with $\sigma _1 p_1 + \dotsc + \sigma _l p_l=0$ is also quite small.) The more complicated task is to show that $Y_{\ell }$ is reasonably equidistributed in arithmetic progressions.

The main issue here is that we have a great number of conditions as in (9) to exclude. Inclusion-exclusion involves $2^m$ terms for m conditions—that is too many. There is a tool for dealing with that sort of issue in number theory, at least in some specific contexts: sieves.

We shall first show how to set up a general, abstract combinatorial sieve, for arbitrary logical conditions (rather than conditions of the form $n\equiv a \bmod p$). We will then show how to apply it to conditions of the form $n\equiv a \bmod m$, that is, congruence conditions where the moduli may be composite (as opposed to being prime, as is common in sieve theory). The matter is tricky—one has to prevent combinatorial explosion again. Rota’s cross-cut theorem will be our friend.

Lastly, we will show how to apply the sieve in our context (with moduli $p p_1 \cdots p_l$ coming from (9) and sketch how to estimate the main and error terms. We will introduce sieve graphs.

For readers who have had some passing contact with sieve theory: while, in introductory texts on sieve theory, the emphasis is often on counting a set of elements S not obeying any of a set of conditions (e.g., the set S of primes n such that $n+2$ is also a prime; its elements n do not fulfill the conditions $n\equiv 0 \bmod p$ or $n\equiv -2 \bmod p$ for any small prime p), the emphasis on much recent work, and also here, lies more generally on providing an approximation to the characteristic function $1_S$ of S by a function that is easier to deal with, or, if you wish, has a “simpler description” (in some precise sense). One label that has become attached to this use of sieves is “envelo** sieve”, though that really describes one kind of approximation (a majorant of $1_S$) and at any rate should really be called an envelo** use of a sieve (many sieves can be used as envelo** sieves). At any rate, that is all more or less orthogonal to the main issue here, which is that we have to develop a genuinely more general sieve.

1.1 An Abstract Combinatorial Sieve

Let $\textbf{Q}$ be a set of conditions that an element x of a set Z may or may not fulfill. (For us, later, Z will be the set of integers, but that is of no importance at this point.) Denote by $\textbf{Q}(x)\in 2^{\textbf{Q}}$ the set $\{Q\in \textbf{Q}: Q(x) \text { is true}\}$, i.e., the set of conditions in $\textbf{Q}$ fulfilled by x. Define $1_\emptyset (S)$ to be 1 if the set S is empty, and 0 otherwise. Then $1_\emptyset (\textbf{Q}(x))$ equals 1 when x satisfies none of the conditions in $\textbf{Q}$, and 0 otherwise.

We are interested in approximations to $1_\emptyset (\textbf{Q}(x))$, i.e., the function that takes the value 1 when x satisfies none of the conditions in $\textbf{Q}$, and 0 otherwise. This may seem to be a silly question, though it falls within the general framework we were discussing before. Let us put matters a little differently. A standard way to express $1_\emptyset (\textbf{Q}(x))$ would be as

$$\begin{aligned} 1_\emptyset (\textbf{Q}(x)) = \sum _{\textbf{T}\subset \textbf{Q}(x)} (-1)^{|\textbf{T}|}, \end{aligned}$$

and that might suit us, except that the number of subsets $\textbf{T}\subset \textbf{Q}(x)$ is very large. Can we obtain a reasonable approximation by means of a sum of the form

$$\begin{aligned} \sum _{\textbf{T}\subset \textbf{Q}(x)} g(\textbf{T}) (-1)^{|\textbf{T}|}, \end{aligned}$$

where $g:2^\textbf{Q}\rightarrow \{0,1\}$ is a function—preferably one whose support is much smaller than $\textbf{Q}(x)$? (Here, as is usual, $2^\textbf{Q}$ denotes the set of all subsets of $\textbf{Q}$.)

It turns out to be possible to bound the error term in an approximation of this form in full generality. To be precise: the error term will be bounded in terms of the boundary of the support of g. Here we say that an $\textbf{S}\subset \textbf{Q}$ is in the boundary of a collection $B\subset 2^\textbf{Q}$ if there is an element s of $\textbf{S}$ such that exactly one of the two sets $\textbf{S}$, $\textbf{S}\setminus \{s\}$ is in B.

Lemma 5

Let $g:2^\textbf{Q}\rightarrow \{0,1\}$. Assume $g(\emptyset )=1$. Choose a linear ordering for $\textbf{Q}$. Then

$$\begin{aligned} 1_\emptyset (\textbf{Q}(x))&= \sum _{\textbf{T}\subset \textbf{Q}(x)} g(\textbf{T}) (-1)^{|\textbf{T}|}\\&+ \mathop {\sum _{\emptyset \ne \textbf{S}\subset \textbf{Q}(x)}}_{\min (\textbf{Q}(x))\in \textbf{S}} (-1)^{|\textbf{S}|} (g(\textbf{S}\setminus \{\min (\textbf{Q}(x))\})-g(\textbf{S})). \end{aligned} $$

The proof is short and basically trivial (for $\textbf{Q}(x)$ non-empty, the second sum is just a reordering of the first sum, with opposite sign). It is inspired by a passage in the proof of Brun’s combinatorial sieve (see, e.g., [1, Sect. 6.2, pp. 87–89]. We do not need the linear ordering for $\textbf{Q}$ to be in any sense natural.

1.2 Sieving by Composite Moduli

Let $\mathscr {Q}$ be a finite collection of arithmetic progressions. To each progression $P\in \mathscr {Q}$, we can associate the condition $n\in P$, for $n\in \mathbb {Z}$. Thus, we obtain a set $\textbf{Q}$ of conditions corresponding to $\mathscr {Q}$, and apply the framework above.

We are interested in approximating $1_{n\not \in P \forall P\in \mathscr {Q}}(n)$—that is, the characteristic function of the set of all n lying in no progression $P\in \mathscr {Q}$—by a sum

$$\begin{aligned} F_\mathfrak {D}(n) = \mathop {\sum _{\mathscr {S}\subset \mathscr {Q}}}_{\bigcap \mathscr {S} \in \mathfrak {D}} (-1)^{|\mathscr {S}|} 1_{n\in \bigcap \mathscr {S}}, \end{aligned}$$

where $\mathfrak {D}\subset \mathscr {Q}^\cap$ is some set of progressions.

We will denote by $\mathfrak {q}(R)$ the modulus q of an arithmetic progression $a + q \mathbb {Z}$.

Proposition 2

Let $\mathscr {Q}$ be a finite collection of distinct arithmetic progressions in $\mathbb {Z}$ with square-free moduli. Let $\mathfrak {D}$ be a non-empty subset of $\mathscr {Q}^\cap = \{\bigcap \mathscr {S}: \mathscr {S}\subset \mathscr {Q}\}$ with $\emptyset \not \in \mathfrak {D}$. Assume $\mathfrak {D}$ is closed under containment, i.e., if $S\in \mathfrak {D}$, then every superset $S'\supset S$ in $\mathscr {Q}^\cap$ is also in $\mathfrak {D}$. Let $F_\mathfrak {D}$ be as above. Then

$$\begin{aligned} 1_{n\not \in P \forall P\in \mathscr {Q}}(n)&= F_{\mathfrak {D}}(n) + O^*\left( \sum _{R\in \partial \mathfrak {D}} 2^{\omega (\textbf{q}(R))} 1_{n\in R}\right) \\&= F_{\mathfrak {D}}(n) + O^*\left( \sum _{R\in \partial _{\textrm{out}} \mathfrak {D}} 3^{\omega (\textbf{q}(R))} 1_{n\in R}\right) \end{aligned}$$

where

$$\partial \mathfrak {D} = \{R\in \mathfrak {D}: \exists P\in \mathscr {Q}\; \text {s.t.}\; P\cap R \not \in \mathfrak {D}\},$$

$$\partial _\textrm{out} \mathfrak {D} = \{D\in \mathscr {Q}^\cap \setminus \mathfrak {D} : \exists P\in \mathscr {Q}, R \in \mathfrak {D}\, \text {s.t.}\, D = P \cap R \}.$$

Moreover, we can write $F_\mathfrak {D}(n)$ in the form

$$\begin{aligned} F_\mathfrak {D}(n) = \sum _{R\in \mathfrak {D}} c_R 1_{n\in R} \end{aligned}$$

with $|c_R|\le 2^{\omega (\mathfrak {q}(R))}$.

We can of course think of $\partial \mathfrak {D}$ and $\partial _{\textrm{out}} \mathfrak {D}$ as the boundary and the outer boundary of $\mathfrak {D}$.

Proof

The proof of the Proposition starts with an application of the Lemma above. In what then follows, the important thing is to prevent a combinatorial explosion. For instance, it is not a priori clear that $c_R$ can be bounded well: there could be very many ways to express a given $R\in \mathfrak {D}$ as an intersection $\bigcap \mathscr {S}$; in fact, the number of ways could be close to $2^{2^{\omega (\mathfrak {q}(R))}}$, that is, the number of collections of subsets of a set with $\mathfrak {q}(R)$ elements. We can give the much better bound $2^{\omega (\mathfrak {q}(R))}$ by obtaining cancellation (by $(-1)^{|\mathscr {S}|}$) among those different ways. To be more precise, we apply the following Lemma, which is an easy consequence of Rota’s cross-cut theorem but can also be proved from scratch in a couple of lines. The same Lemma allows us to deal with the same combinatorial explosion in the error terms.

Lemma 6

Let $\mathscr {C}$ be a collection of subsets of a finite set X. Then

$$\begin{aligned} \left| \mathop {\sum _{\mathscr {S}\subset \mathscr {C}}}_{\bigcup \mathscr {S} = X} (-1)^{|\mathscr {S}|}\right| \le 2^{|X|}. \end{aligned}$$

Proof

Exercise.

How to apply the Proposition? We can define $\mathfrak {D}$ to be the set of progressions in $\mathscr {Q}^\cap$ with “small modulus”, for some notion of “small”. Then its boundary consists of progressions that are “borderline small”, i.e., not really small, and so the proportion of n in each one of them will not be large; we just need to control the size of the boundary to show that the total error term is acceptable.

1.3 Sieve Graphs and Their Usage

We now come to our application of the sieve we have just developed. Our aim is to prevent our walks

$$\begin{aligned} n, n+\sigma _1 p_1, n+\sigma _1 p_1 + \sigma _2 p_2,\dotsc \end{aligned}$$

from having what we called premature revenants. We will do so by constraining each of $n, n+\sigma _1 p_1,n+\sigma _1 p_1 + \sigma _2 p_2\dotsc$ to lie within the set $Y_{\ell }$ of integers that cannot give rise to premature revenants.

To be precise: we define $Y_{\ell }$ to be the set of all integers n except for those for which there are primes $p_1,\dotsc ,p_l\in \textbf{P}$ and signs $\sigma _1,\dotsc ,\sigma _l\in \{-1,1\}$ with $1\le l< \ell$ such that

$$\begin{aligned} p_1|n, p_2|n+\sigma _1 p_1,\dotsc ,p_l|n+\sigma _1 p_1 + \dotsc + \sigma _{l-1} p_{l-1},\end{aligned}$$

(10)

there are no repeated primes among $p_1,\dotsc ,p_l$ except perhaps for consecutive primes $p_i=p_{i+1} = \dotsc =p_j$ with $\sigma _i = \sigma _{i+1} = \dotsc = \sigma _j$, and one of the following two conditions holds:

• there exists a prime $p_0\in \textbf{P}$ distinct from $p_1,\dotsc ,p_l$ such that

  $$\begin{aligned} p_0|n\;\text {and}\;p_0|n+\sigma _1 p_1 + \dotsc + \sigma _l p_l,\end{aligned}$$

  (11)

• we have

  $$\begin{aligned} \sigma _1 p_1 + \dotsc + \sigma _l p_l= 0.\end{aligned}$$

  (12)

The set of integers n that obey conditions (10) and (11) is an arithmetic progression to modulus $[p_0,p_1,\dotsc ,p_l]$ (that is, the lcm of $p_1,\dotsc ,p_l$, i.e., the product of all distinct primes among them), unless it is empty. The set of integers n obeying (10) and (12) is an arithmetic progression to modulus $[p_1,p_2,\dotsc ,p_l]$, unless it is empty. Let $W_{\ell ,\textbf{P}}$ denote the set of all arithmetic progressions arising in this way. Then the condition $n\in Y_\ell$ is equivalent to n not lying in any of the arithmetic progressions in $W_{\ell ,\textbf{P}}$. Likewise, for $\beta _1,\dotsc ,\beta _{2 k}\in \mathbb {Z}$, the condition that $n+\beta _i\in Y_{\ell }$ for all $1\le i\le 2 k$ is equivalent to asking that n not be in any arithmetic progression of the form $P-\beta _i$ with $P\in W_{\ell ,\textbf{P}}$ and $1\le i\le 2 k$.

We are thus in the kind of situation to which our sieve for composite moduli is applicable. Applying the Proposition above, we obtain, for any $m\ge 1$,

$$1_{n+\beta _i\in Y_{\ell } \forall 1\le i\le 2 k}= \mathop {\sum _{R\in \mathscr {Q}^\cap }}_{\omega (\mathfrak {q}(R))\le m} c_R 1_{n\in R} + O^*\left( 3^{m+\ell } \mathop {\sum _{R\in \mathscr {Q}^\cap }}_{m<\omega (\mathfrak {q}(R))\le m+\ell } 1_{n\in R} \right) $$

for $\mathscr {Q} = W_{\ell ,\textbf{P}}(\beta )$ and some $c_R\in \mathbb {R}$ with $|c_R|\le 2^{\omega (\mathfrak {q}(R))}$. (The proof is one line: we define $\mathfrak {D}$ to be the set of all non-empty $R\in \mathscr {Q}^\cap$ such that the modulus of R has $\le m$ prime factors.)

The more obvious issue now is how to bound the error term here. (For us, in our application, there is also the related issue of showing that the main term is well behaved—in particular, its sum over certain arithmetic progressions should not be too large.)

To keep track of the kind of conditions giving rise to a progression $R\in \mathscr {Q}^\cap$, we find it sensible to define a sieve graph, which is, one may say, a pictorial representation of those conditions, or rather of what their general shape is and how they relate to each other.

We define a sieve graph to be a directed graph consisting of:

• 1. a path of length 2k, called the horizontal path;

• 2. threads of length $< \ell$, of two kinds:

  • 1. a closed thread, which is a cycle that contains some vertex of the horizontal path, and is otherwise disjoint from it,

  • 2. an open thread, which is a path that has an endpoint at some vertex of the horizontal path, and is otherwise disjoint from it;

• 3. for each open thread and each of the two endpoints of that thread, an edge whose tail is that endpoint, but whose head belongs only to the edge (i.e., it is a vertex of degree 1). These two edges will be called the thread’s witnesses; they are considered to be part of the thread (Fig. 1).

Fig. 1

A sieve graph: the blue path is the horizontal path, and the witness edges of each open path are in red. Any resemblance to a khipu is both intentional and ahistorical

We will work with pairs $(G,\sim )$, where G is a sieve graph and $\sim$ is an equivalence relation on the edges of G such that the witnesses of a thread are equivalent to each other. We can put additional conditions, reflecting the conditions defining $Y_{\ell }$: we require that witnesses be equivalent to no other edges in their thread, and that, in any thread, the set of edges in an equivalence class form a connected subgraph (meaning: primes do not repeat in a thread unless they are consecutive).

At any rate, it is clear what we will do: we will go over different pairs $(G,\sim )$, and, for each pair, we will consider the divisibility conditions resulting from assigning a distinct prime in $\textbf{P}$ to each equivalence class of $\sim$. We recall that $Y_{\ell }$ is defined as the set of integers that do not satisfy any conditions of a certain kind. A pair $(G,\sim )$, together with an assignment of a prime in $\textbf{P}$ to each equivalence class of $\sim$, corresponds to a conjunction $Q_1\wedge Q_2\wedge \cdots \wedge Q_j$ of some such conditions $Q_i$. (To be precise - the conditions are given by a pair $(G,\sim )$ together with a subset $\textbf{l}\subset \{1,2,\dotsc ,2k\}$, corresponding to the “lit” edges: only those edges in the horizontal path whose indices are in $\textbf{l}$ impose divisibility conditions - the unlit edges are muted, so to speak.) We need to study these conditions $Q_i$ (all of which are of the form “n belongs to an arithmetic progression”, as we have seen) because they will appear in the approximation to $1_{Y_{\ell }}$ that a sieve will give us.

We say $(G,\sim )$ is non-redundant if every thread contains at least one edge x (possibly a witness) whose equivalence class [x] contains no edge in any other thread. (A thread where every edge is equivalent to an edge in some other thread would correspond to a condition that either is redundant, given the conditions from the other threads, or contradicts them. As Caliph Omar did not say...) The cost $\kappa (G,\sim )$ is the number of equivalence classes that contain at least one edge (possibly a witness) in some thread, i.e., the number of classes that do not contain only edges in the horizontal path. Let $\textbf{W}_{k,\ell ,m}$ be the set of non-redundant pairs with given parameters k and $\ell$ and cost m. It is clear that $\textbf{W}_{k,\ell ,m}$ must be finite, since any pair of cost m contains at most m threads. It is not hard to bound the number of elements of $\textbf{W}_{k,\ell ,m}$ with a given number of threads $r\le m$.

When we apply our sieve for composite moduli so as to approximate $1_{n+\beta _i\in Y_\ell \forall 1\le i\le 2k}$, we define $\mathfrak {D}$ to be the set of conditions corresponding to non-redundant pairs $(G,\sim )$ with cost $\kappa (G,\sim )\le m$ for some value of m we choose. Then the outer boundary $\partial _{\textrm{out}} \mathfrak {D}$ corresponds to non-redundant pairs $(G,\sim )$ of cost $m<\kappa (G,\sim )\le m+\ell$. Our task is then to prove that the contribution of all $(G,\sim )$ with cost between m and $m+\ell$ is small. The crucial part is to show that, given any such $(G,\sim )$, together with a sign $\sigma _y$ for each edge y and a subset $\textbf{l}\subset \textbf{k}$, the sum of

$$\prod _{i\in \textbf{k}\setminus \textbf{l}} \frac{1}{p_{[i]}} \prod _{[x]\not \subset \textbf{k}\setminus \textbf{l}} \frac{1}{p_{[x]}} $$

over all choices of $p_{[x]}\in \textbf{P}$ (per equivalence class [x] of $\sim$) is small: it is at most

$$\begin{aligned} \mathscr {L}^{s-r} \left( \frac{\log H}{H_0}\right) ^r, \end{aligned}$$

where r is the number of threads in G and s is the number of equivalence classes of $\sim$. (Recall that $\textbf{P}\in [H_0,H]$ and $\mathscr {L} = \sum _{p\in \textbf{P}} 1/p$.)

The proof is simple, and its main idea is as follows. Since $(G,\sim )$ is non-redundant, each thread contains an edge in an equivalence class that does not appear in other threads (or elsewhere in the same thread, except for consecutive appearances). For each thread, we choose one such edge x. Then the thread binds the variable $p_{[x]}$, so to speak, and so we lose one degree of freedom for each r. In detail:

• 1. for a closed thread, the sum $\sum _y \sigma _y p_{[y]}$ over the edges y of the thread is 0, and so $p_{[x]}$ is determined by the other $p_{[y]}$;

• 2. for an open thread and x a witness, $p_{[x]}$ can range only over the prime divisors of $\sum _y \sigma _y p_{[y]}$;

• 3. for an open thread and x not a witness, given $p_{[y]}$ for the other [y] in the thread, and given $p_{[z]}$ for z the thread’s witness, the class of $p_{[x]}$ modulo $p_{[z]}$ is determined.

We can give estimates on the main term of the sieve in much the same way, only kee** track of the set $\textbf{W}_{k,\ell ,m}'$ of strongly non-redundant pairs, meaning pairs $(G,\sim )$ such that every thread contains at least one edge x (possibly a witness) whose equivalence class [x] contains no edge in any other thread and no lit edge in the horizontal path (that is, no edge in the horizontal path with index in $\textbf{l}$).