Covering systems with restricted divisibility

TL;DR

This paper proves that any covering system with distinct moduli must include a modulus divisible by 2 or 3, revealing a fundamental restriction on the structure of such systems.

Contribution

It establishes a new necessary condition for the existence of distinct covering systems, showing they must involve moduli divisible by 2 or 3.

Findings

Every distinct covering system has a modulus divisible by 2 or 3.

Provides a fundamental restriction on the structure of covering systems.

Advances understanding of divisibility constraints in covering systems.

Abstract

We prove that every distinct covering system has a modulus divisible by either 2 or 3.

Equations

$$a_i\mathrm{mod}{m}_i,i=1,2,...,k$$

$$Q=\mathrm{LCM}(m:m\in \mathcal{M})$$

$$R=\mathbb{Z}\setminus \bigcup _{m\in \mathcal{M}}({\mathbf{a}}_m\mathrm{mod}m),$$

$$({\mathbf{a}}_m\mathrm{mod}m),m\in \mathcal{M}$$

$$\mathbf{P}(R)=\mathbf{P}\left(\bigcap _{m|Q}{A}_m^c\right).$$

$$0\le \mathbf{P}(A_T)\le \prod _{t\in T}{\pi }_t:={\pi }_T.$$

$$\rho (T)=1-\sum _{\varnothing \ne S_1\subset T}{\pi }_{S_1}+\sum _{\begin{array}{c}\varnothing \ne S_1,S_2\subset T\\ S_1<S_2\text{ disjoint}\end{array}}{\pi }_{S_1}{\pi }_{S_2}-\sum _{\begin{array}{c}\varnothing \ne S_1,S_2,S_3\subset T\\ S_1<S_2<S_3\text{ disjoint}\end{array}}{\pi }_{S_1}{\pi }_{S_2}{\pi }_{S_3}+....$$

$$\mathbf{P}\left(\bigcap _{\varnothing \ne S\subset T}{A}_S^c\right)\ge \rho (T)>0$$

$$\frac{\mathbf{P}\left({\bigcap }_{\varnothing \ne S\subset T_2}{A}_S^c\right)}{\mathbf{P}\left({\bigcap }_{\varnothing \ne S\subset T_1}{A}_S^c\right)}\ge \frac{\rho (T_2)}{\rho (T_1)}.$$

$$A_S=\bigcup _{m:\mathrm{sqf}(m)=m_S}(a_m\mathrm{mod}m),$$

$$\mathbf{P}(A_S)<\prod _{i\in S}\frac{1}{p_i-1}.$$

$$\frac{\mathbf{P}((b\mathrm{mod}m)\cap R)}{\mathbf{P}(R)}$$

$$\le \mathbf{P}((b\mathrm{mod}m))\frac{\mathbf{P}\left({\bigcap }_{m^{\prime }\in \mathcal{M},m^{\prime }|Q,(m,m^{\prime })=1}(a_{m^{\prime }}\mathrm{mod}{m}^{\prime }{)}^c\right)}{\mathbf{P}\left({\bigcap }_{m^{\prime }\in \mathcal{M},m^{\prime }|Q}(a_{m^{\prime }}\mathrm{mod}{m}^{\prime }{)}^c\right)}$$

$$=\frac{1}{m}\frac{\mathbf{P}\left({\bigcap }_{m^{\prime }\in \mathcal{M},m^{\prime }|Q,(m,m^{\prime })=1}(a_{m^{\prime }}\mathrm{mod}{m}^{\prime }{)}^c\right)}{\mathbf{P}\left({\bigcap }_{m^{\prime }\in \mathcal{M},m^{\prime }|Q}(a_{m^{\prime }}\mathrm{mod}{m}^{\prime }{)}^c\right)}.$$

$$\frac{\mathbf{P}((b\mathrm{mod}m)\cap R)}{\mathbf{P}(R)}\le \frac{1}{m}\frac{\rho ([n]\setminus S_m)}{\rho ([n])}.$$

$$\underset{b\mathrm{mod}m}{\max }\frac{|R\cap (b\mathrm{mod}m)|}{|R|}\le \frac{1}{m}\frac{\rho (q_{[n]\setminus S})}{\rho (q_{[n]})}.$$

$$R=\mathbb{Z}\setminus \bigcup _{n\in \mathcal{N}}({\mathbf{a}}_n\mathrm{mod}n).$$

$$\forall p\in \mathcal{P},x_p\ge \sum _{n\in \mathcal{N}:p|n}\frac{|{\mathbf{a}}_n\mathrm{mod}n|{\prod }_{p^{\prime }|n}(1+x_{p^{\prime }})}{n}.$$

$$\frac{|R\mathrm{mod}Q|}{Q}\ge \exp \left(-\sum _{n\in \mathcal{N}}\frac{|{\mathbf{a}}_n\mathrm{mod}n|{\prod }_{p|n}(1+x_p)}{n}\right)>0.$$

$$\underset{b\mathrm{mod}n}{\max }\frac{|R\cap (b\mathrm{mod}n)\mathrm{mod}Q|}{|R\mathrm{mod}Q|}\le \frac{\exp \left({\sum }_{p|n}{x}_p\right)}{n}.$$

$$G_p(\underline{x})=\sum _{n\in \mathcal{N}:p|n}\frac{|{\mathbf{a}}_n\mathrm{mod}n|{\prod }_{p^{\prime }|n}(1+x_{p^{\prime }})}{n}$$

$$\mathcal{M}\subset \{m>1,(m,6)=1\},$$

$$Q=\mathrm{LCM}(m:m\in \mathcal{M}),$$

$$R=\mathbb{Z}\setminus \bigcup _{m\in \mathcal{M}}(a_m\mathrm{mod}m)$$

$$Q_i=\prod _{p^j\Vert Q,p<P_i}{p}^j$$

$${\mathcal{N}}_i=\{n>1:n|Q_i,p|n\Rightarrow P_{i-1}<p\le P_i\}.$$

$$\forall i\ge 1,R_i=\mathbb{Z}\setminus \bigcup _{m\in {\mathcal{M}}_i}(a_m\mathrm{mod}m).$$

$$R_{i+1}=R_i\setminus \bigcup _{m\in {\mathcal{M}}_{i+1}\setminus {\mathcal{M}}_i}(a_m\mathrm{mod}m),$$

$$R_{i+1}\cap (r\mathrm{mod}{Q}_i)=(r\mathrm{mod}{Q}_i)\setminus \bigcup _{n\in {\mathcal{N}}_{i+1}}{A}_{n,r},$$

$$A_{n,r}=(r\mathrm{mod}{Q}_i)\cap \bigcup _{\begin{array}{c}{m}_0|Q_i\\ m_{0}n\in {\mathcal{M}}_{i+1}\end{array}}(a_{m_{0}n}\mathrm{mod}{m}_{0}n).$$

$$\forall r\in R_0^{\ast }\cap R_1\mathrm{mod}{Q}_1,{\mu }_1(r)=\frac{1}{|R_1\mathrm{mod}{Q}_1|}.$$

Full text

The project was sponsored by the National Security Agency under Grant Number H98230-16-1-0048 and by the National Science Foundation under grant numbers DMS-1712682 and DMS-1802336.

1. Introduction

A covering system of congruences is a collection

[TABLE]

such that every integer satisfies at least one of them. A covering system is distinct if the moduli $m_{i}$ are distinct and greater than 1. Erdős introduced the idea of a distinct covering system of congruences in constructing an arithmetic progression of odd numbers, none of whose members are prime [3]. In the paper [3] Erdős asked whether the least modulus of a distinct covering system of congruences can be arbitrarily large. The first author recently answered this question in the negative [8], proving that the least modulus of a distinct covering system of congruences is at most $10^{16}$. The largest known minimum modulus is 42, given by Tyler Owens [10]. A second old problem of Erdős and Selfridge asks whether there exists a distinct covering system of congruences with all moduli odd. According to [4] Erdős has offered $25 for the proof that no odd distinct covering system of congruences exists, while Selfridge has offered$2000 for a construction of an odd distinct covering system. Schinzel proved that a negative answer to the odd modulus problem has applications to the irreducibility of families of polynomials. While the odd modulus problem remains open, Simpson and Zeilberger [14] proved that a distinct covering system consisting of odd square-free numbers involves at least 18 primes, which was improved to 22 primes by Guo and Sun [6]. This paper makes further negative progress towards the odd modulus problem.

Theorem 1**.**

Every distinct covering system of congruences has a modulus divisible by either $2$ or $3$.

This answers a problem raised in [7].

2. Set-up

Suppose given a finite set of moduli $\mathscr{M}$, and, for each $m\in\mathscr{M}$, a set of residues $\mathbf{a}_{m}$ modulo $m$. Let

[TABLE]

and

[TABLE]

which is a set defined modulo $Q$. One way to show that the congruences

[TABLE]

do not cover the integers is to give a positive lower bound for the density of $R$. The proof of Theorem 1 gives such a lower bound, although quantitatively it estimates some related quantities.

If we let $\mathbb{Z}/Q\mathbb{Z}$ have the uniform probability measure, then the density of $R$ is equal to its probability. For $m\in\mathscr{M}$ let $A_{m}$ be the event $(\mathbf{a}_{m}\bmod m)$, which has probability $\frac{|\mathbf{a}_{m}|}{m}$, and extend this to $m|Q$ with $m\not\in\mathscr{M}$ by setting $A_{m}=\emptyset$ for these $m$. Then

[TABLE]

A familiar argument (the Chinese Remainder Theorem) implies that $A_{m}$ is independent of any set of congruences to moduli co-prime to $m$. Thus a valid dependency graph for the events $\{A_{m}:m|Q\}$ has edge $(m,m^{\prime})$ if and only if $\operatorname{GCD}(m,m^{\prime})>1$.

A family of results connected to the Lovász Local Lemma give worst-case lower bounds for the probability of an intersection as in (1), taking as input only the events’ probabilities and their dependency graph. In principle we could hope to prove Theorem 1 by directly applying one of these results to claim that the uncovered set always has a non-zero density, but, as we will see, such a lower bound cannot be given, and further input is needed. Two methods of Lovász type do figure into our argument, however, as we will describe.

Given the problem of estimating from below the probability of the intersection of the complements of some events given only their probabilities and their dependency graph, the best possible estimate has been given by Shearer [13]. The estimate is best possible in the sense that the argument constructs a probability space and events having the prescribed probabilities and dependency graph, and such that the lower bound holds with equality. However, the condition with which Shearer’s result holds can be difficult to verify, and so the following result is useful because it is easy to check. Note that this is essentially due to [14] in this context.

Theorem 2** (Shearer-type theorem).**

Suppose we have a probability space. Let $[n]=\{1,2,...,n\}$, and assume that for each $1\leq i\leq n$ there is a weight $\pi_{i}$ assigned, satisfying $1\geq\pi_{1}\geq\pi_{2}\geq...\geq\pi_{n}\geq 0$. Let the sets $\emptyset\neq T\subset[n]$ index events $A_{T}$ each having probability

[TABLE]

Assume that $A_{T}$ is independent of the $\sigma$-algebra generated by $\{A_{S}:S\subset[n],S\cap T=\emptyset\}$, so that a valid dependency graph for the events $\{A_{T}:\emptyset\neq T\subset[n]\}$ has an edge between $S\neq T$ whenever $S\cap T\neq\emptyset$.

Define $\rho(\emptyset)=1$, and given $\emptyset\neq T\subset[n]$, set (put an arbitrary total ordering $<$ on $2^{[n]}$ to avoid confusion)

[TABLE]

Suppose that $\rho([1])\geq\rho([2])\geq...\geq\rho([n])>0.$ Then for any $\emptyset\neq T\subset[n]$,

[TABLE]

and, for any $T_{1}\subset T_{2}\subset[n]$,

[TABLE]

We prove a slightly more general version of this theorem in Appendix C.

To apply the Shearer-type theorem in the context of Theorem 1, order the primes greater than 3 as $p_{1}=5,p_{2}=7,p_{3}=11,...$. Suppose we are given a distinct congruence system with moduli formed with the primes $p_{1},...,p_{n}$. Identify $S\subset[n]$ with the square-free number $m_{S}=\prod_{i\in S}p_{i}$ and form the event $A_{S}$ which is the union of all congruences having square-free part $m_{S}$,

[TABLE]

where $\operatorname{sqf}(m)=\prod_{p:p|m}p$. Then $A_{S}$ is an event with probability

[TABLE]

In particular, we may appeal to Theorem 2 with $\pi_{i}=\frac{1}{p_{i}-1}$. Arguing in this way, we may check that there is no covering composed of only the primes between 5 and 631, but at this point, the Shearer function becomes negative, and no further result can be drawn from that estimate.

What allows us to make further progress is that, within the range in which Shearer’s theorem holds, estimate (3) of Theorem 2 gives substantial information about the structure of the uncovered set. To see this, suppose that we have a congruence system as above with uncovered set $R$, and that Theorem 2 applies. We can estimate the proportion of the set $R$ that lies in a given congruence class $(b\bmod m)$ for $m|Q$ by

[TABLE]

The ratio of probabilities on the right is bounded by the relative conclusion (3) of Theorem 2, which gives a ratio of $\frac{\rho([n]\setminus S_{m})}{\rho([n])}$ where $[n]$ again represents the full set of primes dividing $Q$, and $S_{m}$ is those primes from $[n]$ which divide $m$. Thus

[TABLE]

If $S_{m}$ is such that $\rho([n]\setminus S_{m})\approx\rho([n])$ then we deduce that $R$ is almost uniformly distributed across residues modulo $m$.

We summarize the above discussion in the following Theorem.

Theorem 3**.**

Let $p_{1}<p_{2}<...<p_{n}$ be a sequence of primes, and let weights $\pi_{1},...,\pi_{n}$ given by $\pi_{i}=\frac{1}{p_{i}-1}$. For a subset $S\subset[n]$ identify $S$ with $q_{S}=\prod_{p\in S}p,$ and write $\rho(q)=\rho(q_{S})=\rho(S)$ for the Shearer function associated to $S$ with weights $\pi_{i}$, as in Theorem 2.

Suppose that $\rho(p_{1})\geq\rho(p_{1}p_{2})\geq...\geq\rho(p_{1}p_{2}...p_{n})>0$. Then any distinct congruence system with moduli composed only of $p_{1},...,p_{n}$ does not cover the integers. Moreover, if $R$ is the uncovered set and if $m$ is a modulus composed of primes corresponding to a set $S\subset[n]$ then

[TABLE]

Although the sieving problem described in Theorem 1 concerns systems of congruences in which each congruence set $\mathbf{a}_{m}$ has size 0 or 1, in the course of our argument we consider congruences with sets $\mathbf{a}_{m}$ of variable size. In this situation the condition of Theorem 2 becomes unwieldy and we appeal instead to the following Theorem, which follows from an improved form of the Lovász Local Lemma due to [1], see also [12].

Theorem 4**.**

Let $\mathscr{N}\subset\mathbb{N}_{>1}$ be a finite collection of moduli whose prime factors are drawn from a set of primes $\mathscr{P}$. Let $\operatorname{LCM}(n:n\in\mathscr{N})=Q$. Suppose that for each $n\in\mathscr{N}$ a collection of residues $\mathbf{a}_{n}\bmod n$ is given. Write

[TABLE]

Suppose that there exist weights $\{x_{p}\}_{p\in\mathscr{P}}$ with $x_{p}\geq 0$, which satisfy the constraints

[TABLE]

Then the density of $R$ is at least

[TABLE]

Also, for any $n\in\mathscr{N}$,

[TABLE]

Remark*.*

Conclusion (5) corresponds to (2) of Theorem 2, and (6) corresponds to (4).

If we write $\underline{x}$ for $\{x_{p}\}_{p\in\mathscr{P}}$ and $\underline{G}(\underline{x})$ for

[TABLE]

then the condition of Theorem 4 equivalently asks for a non-negative ($\underline{x}\geq\underline{0}$) fixed point $\underline{G}(\underline{x})=\underline{x}$, which is relatively easy to determine. Thus, although Theorem 4 is strictly weaker than Theorem 2, it is useful since it is more easily applied.

A proof and further discussion of Theorem 4 is given in Section 4.

3. Overview of argument

We now give an overview of our argument. As the structure is similar to that of the minimum modulus problem we refer the proofs of some background statements to [8].

We assume given a congruence system with finite set of moduli

[TABLE]

together with a residue class $a_{m}\bmod m$ for each $m\in\mathscr{M}$. We let

[TABLE]

and set

[TABLE]

for the set left uncovered by the congruence system. Theorem 1 follows by showing that the density of $R$ is positive.

To estimate the density of $R$ we appeal to Lovász Local Lemma-type arguments of the previous section. These arguments, however, only apply to estimate the density of sets left uncovered by congruence systems whose moduli are composed of a limited number of primes, and so we break the estimate for the density of $R$ into stages.

Let $P_{0}=4<P_{1}<P_{2}<...$ be a sequence of real numbers (not equal to prime integers). Let $Q_{0}=1$ and, for $i\geq 1$,

[TABLE]

be the part of $Q$ composed of primes less than $P_{i}$. We let $\mathscr{M}_{i}=\{m\in\mathscr{M}:m|Q_{i}\}$ be the $P_{i}$-smooth moduli in $\mathscr{M}$, and we let the set of ‘new factors’ be

[TABLE]

Notice that each $m\in\mathscr{M}_{i+1}\setminus\mathscr{M}_{i}$ has a unique factorization as $m=m_{0}n$ with $m_{0}|Q_{i}$ and $n\in\mathscr{N}_{i+1}$.

We consider the sequence of sets $\mathbb{Z}=R_{0}\supset R_{1}\supset...$,

[TABLE]

Since $R_{i}=R$ eventually, it will suffice to show that $R_{i}$ is non-empty for each $i$.

The set $R_{i}$ is defined modulo $Q_{i}$. Viewing $\mathbb{Z}/Q_{i+1}\mathbb{Z}$ as fibered over $\mathbb{Z}/Q_{i}\mathbb{Z}$ we note that

[TABLE]

so that we may view $R_{i+1}$ as cut out from the fibers $(r\bmod Q_{i})$, $r\in R_{i}$, by congruences to moduli in $\mathscr{M}_{i+1}\setminus\mathscr{M}_{i}$. Given $r\in R_{i}$ and $m\in\mathscr{M}_{i+1}\setminus\mathscr{M}_{i}$, factor $m=m_{0}n$ with $m_{0}|Q_{i}$ and $n\in\mathscr{N}_{i+1}$. Then the congruence $(a_{m}\bmod m)$ meets $(r\bmod Q_{i})$ if and only if $r\equiv a_{m_{0}n}\bmod m_{0}$, and when it does so, it intersects in a single residue class modulo $nQ_{i}$. Thus, grouping together moduli according to common new factor $n\in\mathscr{N}_{i+1}$ we find

[TABLE]

with

[TABLE]

After translating and dilating $(r\bmod Q_{i})$ to coincide with the integers, the set $A_{n,r}$ is composed of some residue classes modulo $n$, a set which we call $\mathbf{a}_{n,r}$. Thus we can understand the problem of estimating the density of $R_{i+1}$ within $(r\bmod Q_{i})$ as sieving the integers by multiple residue classes to moduli in $\mathscr{N}_{i+1}$, a set of moduli whose prime factors are constrained to lie in $[P_{i},P_{i+1})$. This is the situation treated by the Lovász-type Theorem, Theorem 4 above, and so, if we are able to solve the relevant fixed-point problem then we obtain that the fiber is non-empty. Note that in the initial stage, all of the sieving sets have size 0 or 1, so that in this stage we can appeal to the optimal Shearer-type Theorem, Theorem 2.

In practice we will not estimate the density of $R_{i+1}$ over all of $R_{i}$, but only within certain ‘good’ fibers above a subset $R_{i}^{*}\subset R_{i}\bmod Q_{i}$. We will be deliberately vague at this point about the requirements of a good fiber. Roughly these ensure that the corresponding fixed-point problem has a favorable solution. Also, we require that $R_{i}^{*}\subset R_{i-1}^{*}\cap R_{i}$ so that the good sets are nested. We let $R_{0}^{*}=R_{0}=\mathbb{Z}$.

For $i\geq 1$ we weight the set $\mathbb{Z}/Q_{i}\mathbb{Z}$ with a probability measure $\mu_{i}$ supported on $R_{i-1}^{*}\cap R_{i}$, chosen so as to guarantee that a large proportion of the fibers are good. The measure $\mu_{1}$ is uniform on the set $R_{0}^{*}\cap R_{1}=R_{1}\subset\mathbb{Z}/Q_{1}\mathbb{Z}$,

[TABLE]

Taking the measure $\mu_{i}$ as given, define, for $i\geq 1$,

[TABLE]

to be the proportion of good fibers. For $i\geq 1$ and $r\in R_{i}^{*}\cap R_{i+1}\bmod Q_{i+1}$ we set

[TABLE]

Thus, for a fixed $r\in R_{i}$, $\mu_{i+1}$ is constant on $R_{i+1}\cap(r\bmod Q_{i})$. That $\mu_{i}$ is a sequence of probability measures follows from [8] Lemma 2, although, note that the factor of $\frac{1}{\pi_{\operatorname{good}}(i)}$ is not included in the definition of $\mu_{i}$ in [8], so that the measures there do not have mass 1. Throughout, when we write $\mathbf{E}_{r\in R_{i-1}^{*}\cap R_{i}}$ we mean expectation with respect to the measure $\mu_{i}$.

Along with the measure $\mu_{i}$ we track some bias statistics of $R_{i-1}^{*}\cap R_{i}$. Let $\ell_{k}(m)$ be the multiplicative function given at primes powers by

[TABLE]

For $i\geq 1$, the $k$th bias statistic of $R_{i-1}^{*}\cap R_{i}$ is defined to be

[TABLE]

The importance of the bias statistics is that they control moments of (mixtures of) the sizes of the sets $\mathbf{a}_{n,r}$ as $r$ varies in $R_{i-1}^{*}\cap R_{i}$.

Lemma 5**.**

Let $i\geq 1$. Let $\{w_{n}:n\in\mathscr{N}_{i+1}\}$ be any collection of non-negative weights, not all of which are zero. For each $k\geq 1$ we have

[TABLE]

Proof.

See Lemmas 4 and 5 of [8]. ∎

In addition to the bias statistics, it will be useful for us to track maximum biases among the various good fibers. Let $i\geq 0$ and let $n\in\mathscr{N}_{i+1}$. We define the maximum bias at $n$ to be

[TABLE]

Note that these appeared only implicitly in [8], but to get a better quantitative bound it will be useful for us to track them more carefully here.

The iterative growth of the bias statistics $\beta_{k}(i)$ to $\beta_{k}(i+1)$ is controlled by the proportion of good fibers $\pi_{\operatorname{good}}(i)$ and the maximal biases at $n\in\mathscr{N}_{i+1}$.

Lemma 6**.**

Let $i\geq 1$. For each $k\geq 1$ we have the bound

[TABLE]

Proof.

This follows by tracing the proof of Proposition 3 of [8]. ∎

We now turn to giving a detailed account of Theorem 4.

4. The Local Lemma and good fibers

Our Theorem 4, which is used to estimate the density of good fibers, is derived from the following improved version of the Lovász Local Lemma due to [1], see also [12].

Theorem 7** (Clique Lovász Local Lemma).**

Suppose that $G=(V,E)$ is a dependency graph for family of events $\{A_{v}\}_{v\in V}$, each with probability $\mathbf{P}(A_{v})\leq\pi_{v}$. Let $N_{v}$ be the neighborhood of $v\in V$. Suppose that there exists sequence $\underline{\mu}=\{\mu_{v}\}_{v\in V}$ of reals in $[0,\infty)$ such that, for each $v\in V$,

[TABLE]

where

[TABLE]

Then

[TABLE]

and, for all $U\subset V$,

[TABLE]

Remark*.*

In the definition of $\phi_{v}$, $R=\emptyset$ is to be included, with associated product equal to 1.

Proof.

This theorem with conclusion

[TABLE]

is proven in [1], and the corresponding relative conclusion

[TABLE]

follows directly from the argument there. To deduce (8) and (9), observe that

[TABLE]

so that

[TABLE]

∎

Recall that Theorem 4 applies in the context of a congruence system to moduli in a set $\mathscr{N}$, whose prime factors lie in a set $\mathscr{P}$. Each modulus $n\in\mathscr{N}$ has a set of residues $\mathbf{a}_{n}$, considered to be a probabilistic event with probability $\frac{|\mathbf{a}_{n}|}{n}$. We require a system of non-negative weights $\{x_{p}\}_{p\in\mathscr{P}}$ satisfying

[TABLE]

and the conclusion is that the uncovered set $R$ has density at least

[TABLE]

and that, for any $n\in\mathscr{N}$, for any $b\bmod n$,

[TABLE]

Deduction of Theorem 4.

To deduce Theorem 4 from Theorem 7 we take $V$ to be the set of non-trivial square-free products of primes in $\mathscr{P}$,

[TABLE]

The event associated to $v\in V$ is the union of congruences $(\mathbf{a}_{n}\bmod n)$ for which $\operatorname{sqf}(n)=v$, and this event has probability

[TABLE]

The dependency graph connects $v_{1}$ and $v_{2}$ if and only if $\operatorname{GCD}(v_{1},v_{2})>1$.

We take the weight $\mu_{v}$ to be multiplicative, $\mu_{v}=\pi_{v}\prod_{p|v}(1+x_{p})$. This has the effect of reducing (7) at $v$ to the constraint

[TABLE]

Notice that

[TABLE]

since each term in the sum on the left appears in the expansion of the product on the right. Thus if we make the condition that for each $p|Q^{\prime}$,

[TABLE]

which is the condition (12) in the case $v=p$, then (12) holds automatically for all $v$. In this way we have reduced to guaranteeing the system of prime constraints

[TABLE]

which is the constraint of Theorem 4.

The first conclusion, (8) of Theorem 7 now gives that

[TABLE]

which is the first conclusion of Theorem 4. To get the second conclusion, use

[TABLE]

The last term is bounded by

[TABLE]

∎

We now give a sufficient criterion to guarantee a good solution to the fixed point equation governing existence of weights in Theorem 4. Recall that we define

[TABLE]

A trivial lower bound for a fixed point $\underline{G}(\underline{x}^{\operatorname{fix}})=\underline{x}^{\operatorname{fix}}$ is

[TABLE]

and we wish to say that a fixed point lies near $\underline{x}^{0}$. The $n$th derivative $D^{n}\underline{G}(\underline{0})$ is a multilinear map $\bigotimes^{n}\ell^{2}(\mathscr{P})\to\ell^{2}(\mathscr{P})$. Give it the usual operator norm,

[TABLE]

The following theorem guarantees that there exists such a fixed point $\underline{x}^{\operatorname{fix}}$ close to $\underline{x}^{0}$ when there is good control of the operator norms of the derivatives of $D^{n}(\underline{G})(\underline{0})$ of $\underline{G}$ at $\underline{0}$. The theorem was motivated by the series of approximations made in Newton’s method.

Theorem 8**.**

With the notation as above, let $M>0$ be a parameter. Assume that

[TABLE]

and set $B_{2,0}=\|\underline{x}^{0}\|_{\ell^{2}}$ and

[TABLE]

Suppose that $\theta=\frac{B_{\operatorname{op}}}{1-B_{\infty}}<1$ and that $\frac{B_{2,0}}{1-\theta}\leq M$. Then there exists $\underline{x}^{\operatorname{fix}}=\underline{x}^{0}+\underline{\epsilon}$, $\underline{\epsilon}\geq\underline{0}$ solving the fixed point equation $\underline{G}(\underline{x}^{\operatorname{fix}})=\underline{x}^{\operatorname{fix}}$, such that

[TABLE]

Proof.

Let $F(\underline{x})=\underline{G}(\underline{x})-\underline{x}$ so that we seek to solve $F(\underline{x}^{\operatorname{fix}})=\underline{0}$. This we can attempt via ‘Newton’s method’.

Let

[TABLE]

Starting from the initial guess $\underline{x}^{0}$ as above, set $\underline{x}^{i+1}=\underline{x}^{i}+(I-\mathscr{D})^{-1}F(\underline{x}^{i})$. We may also set $\underline{x}^{-1}=\underline{0}$, which is consistent with this definition. A moment’s thought shows that the sequence $\underline{x}^{i}$ is increasing, so that if it is bounded it converges to the desired fixed point, and $\underline{\epsilon}=\sum_{i=0}^{\infty}(\underline{x}^{i+1}-\underline{x}^{i})$. Plainly $\|(I-\mathscr{D})^{-1}\|_{\operatorname{op}}=\frac{1}{1-B_{\infty}}$, so that

[TABLE]

Note that $F$ is a polynomial. Thus

[TABLE]

In this sum, write $DF(\underline{0})=(\mathscr{D}-I)+\mathscr{OD}$, and recall that $\underline{x}^{i}=\underline{x}^{i-1}+(I-\mathscr{D})^{-1}F(\underline{x}^{i-1})$, so that

[TABLE]

On Taylor expanding $F(\underline{x}^{i-1})$ we find

[TABLE]

Now we impose the constraint $\|\underline{x}^{j}\|_{\ell^{2}}\leq M$, which holds for $j=1$, and which we will verify for all $j$ by induction. With this assumption, by the usual trick with the triangle inequality in which we change one coordinate at a time,

[TABLE]

so that $|F(\underline{x}^{i})\|_{\ell^{2}}\leq B_{\operatorname{op}}\|\underline{x}^{i}-\underline{x}^{i-1}\|$ and

[TABLE]

Since

[TABLE]

we have $\|\underline{x}^{i}\|\leq\frac{B_{2,0}}{1-\theta}\leq M$ for all $i$, which verifies the condition above. It follows that $\|\underline{\epsilon}\|_{\ell^{2}}\leq\frac{B_{2,0}\theta}{1-\theta}$. ∎

4.1. Random Lovász weights

For each $i=1,2,...$ on (a subset of good) fibers above $R_{i-1}^{*}\cap R_{i}$ we apply Theorem 4 with moduli $\mathscr{N}=\mathscr{N}_{i+1}$ and residues $\mathbf{a}_{n}=\mathbf{a}_{n,r}$. Thus we think of the quantities from Theorems 4 and 8 as depending upon the random variable $r$, e.g. $\underline{G}(\underline{0})=\underline{G}(\underline{0},r)$, $B_{\infty}=B_{\infty,r}$, $\underline{x}^{0}=\underline{x}^{0}(r)$. We wish to understand properties of the distribution of $\underline{x}^{\operatorname{fix}}(r)$, but will instead define good fibers in terms of $\ell^{p}$ control of $\underline{G}(\underline{0},r)$, and control of $B_{\operatorname{op}}(M,r)$, the other quantities of interest being controlled in terms of these. We now work to control $B_{\operatorname{op}}$.

We directly verify that the partial derivatives of $D^{k}G$ are given by

[TABLE]

A simple bound for the operator norm of $D^{k}\underline{G}(\underline{0})$ is

[TABLE]

which, in view of the evaluation of $D^{k}G$, is given by, for $k\geq 2$,

[TABLE]

In $\mathscr{OD}=D\underline{G}(\underline{0})-\operatorname{diag}(D\underline{G}(\underline{0}))$ the diagonal terms $p=p_{1}$ are missing, so that we recover the bound

[TABLE]

By Cauchy-Schwarz, for positive weights $W_{1},W_{2},W_{3},...$

[TABLE]

Let $\min(\mathscr{P})\geq P+1$. We choose

[TABLE]

from which it follows

[TABLE]

We record bounds for $\|\underline{G}(\underline{0})\|_{\ell^{2}}$ and $S_{1},S_{2},...$ etc averaged over $r\in R_{i-1}^{*}\cap R_{i}$.

Lemma 9**.**

Let $i\geq 1$. For $r\in R_{i-1}^{*}\cap R_{i}$ consider $\mathscr{N}=\mathscr{N}_{i+1}$ and $\mathbf{a}_{n}=\mathbf{a}_{n,r}$ as in the discussion above. We have the following bounds.

[TABLE]

and, for $k\geq 2$,

[TABLE]

Proof.

These follow directly from the convexity lemma, Lemma 5, and the bound, for distinct $P_{i}\leq p_{1},...,p_{k}<P_{i+1}$,

[TABLE]

∎

Inserting (15) in the last lemma, we conclude the following bound.

Lemma 10**.**

Let $B_{\operatorname{op}}$ be the constant from Theorem 8. Averaged over $R_{i-1}^{*}\cap R_{i}$, we have the bound

[TABLE]

with $\mathscr{C}_{i}$ given as above by

[TABLE]

We conclude this section with a brief discussion of how we apply Theorem 4. Beyond demonstrating that fibers above a good set $R_{i}^{*}$ are non-empty, the information that we wish to obtain from Theorem 4 is a bound for the bias statistics $\beta_{k}(i+1)$ in the next stage of iteration. Lemma 6 reduces this problem to bounding the individual biases $b_{n}$ of $R_{i}^{*}\cap R_{i+1}$ at $n\in\mathscr{N}_{i+1}$, and Theorem 4 demonstrates that this bias is bounded by

[TABLE]

We bound this quantity in terms of the number of prime factors $\omega=\omega(n)$ of $n$. Thinking of $\underline{\epsilon}=\underline{\epsilon}(r)$ as a small error, we have

[TABLE]

where $\|\cdot\|_{\infty,k}$ denotes the norm

[TABLE]

Since we typically have information regarding $\|\underline{G}(\underline{0})\|_{\ell^{p}}$ for $p=2$ or 3 (or both) we are led to a maximization problem of the type,

[TABLE]

In the case $p=2$ this may be easily solved along the following lines. It is no loss to assume that $\underline{G}(\underline{0})\in\mathbb{R}_{\geq 0}^{\omega}$. An application of Lagrange multipliers gives that the coordinates of the optimum take at most 3 values, $0=c_{1}<c_{2}\leq\frac{1}{3}\leq c_{3}\leq B_{2}$, subject to $c_{2}(1-c_{2})^{2}=c_{3}(1-c_{3})^{2}$. When there are two non-zero values, $c_{2}$ is constrained by $c_{2}(1-c_{2})^{2}\geq B_{2}(1-B_{2})^{2}$, which is only possible for a bounded number of non-zero entries. For large $\omega$, the optimum is $\frac{\sqrt{\omega}B_{2}}{1-\frac{B_{2}}{\sqrt{\omega}}}$, so that the best choice for all $\omega$ is a finite check.

The case $p=3$ is actually simpler, because, in that case also there are at most 2 non-zero values, and they necessarily satisfy $c_{2}=1-c_{3}$.

5. Explicit calculation in initial stages

In the initial stage, we appeal to the Shearer-type Theorem, Theorem 3, with the primes in the range $P_{0}=4<p<P_{1}=222$ and we verify numerically that the condition of the theorem holds. We also calculate the bound for bias statistics

[TABLE]

The method of performing these explicit computations is described in Appendix A. Empirically, the barrier to ruling out an odd covering using the current method is that the optimal application in the initial stage can only accommodate a few primes, so that the resulting bounds for moments do not permit the process to continue.

Let $P_{2}=4000$.

In order to choose the good set $R_{1}^{*}\subset R_{1}(=R_{0}^{*}\cap R_{1})$ we appeal to Lemmas 9 and 10 to calculate, for any $C_{2},C_{\operatorname{op}}>0$, and for $M=1.769746269$,

[TABLE]

and

[TABLE]

We calculate numerically that

[TABLE]

We choose $C_{2}=\frac{0.9}{0.246514091}$ and $C_{\operatorname{op}}=\frac{0.1}{0.002220166}$, so that the above inequality reads

[TABLE]

We say that $r\in R_{1}$ is good if

[TABLE]

By Markov’s inequality, $\pi_{\operatorname{good}}(1)\geq 0.3$.

Evidently, for all $r$,

[TABLE]

We save a little extra ground by conditioning on the actual size of $\|\underline{G}(\underline{0},r)\|_{\ell^{2}}$. Let $K=100$ be a parameter. For $1\leq j\leq K$ we say that $r\in R_{1}^{*}$ is in bin $\mathscr{B}_{j}$ if

[TABLE]

For $r\in\mathscr{B}_{j}$ we have

[TABLE]

Abusing notation, for $r\in\mathscr{B}_{j}$ we write quantities depending upon $r$ as depending upon $j$ instead so, we use $\underline{G}(\underline{0},j)$, $B_{2}(j)$, $B_{\operatorname{op}}(j)$, and so forth.

In each bin we update

[TABLE]

and thus

[TABLE]

and $\theta(j)\leq\frac{B_{\operatorname{op}}(j)}{1-B_{2}(j)}.$ We check numerically, bin-by-bin, that for all bins,

[TABLE]

so that the condition of Theorem 8 is met. In particular, each good fiber is non-empty.

Again, we apply Theorem 8 bin-by-bin so that, in each bin we obtain a bound of

[TABLE]

Beginning from the information $B_{2}^{2}(j)\leq\frac{j}{K}0.625533539$, we solve the optimization problem (17) for $\omega=1,2,3,...$. As we have already commented, for each $\omega$ the optimal solution has no more than two non-zero values among the $x_{p}$. When it has the two values $c_{1},c_{2}$ these satisfy for some positive integers $a\leq b$, $a+b\leq\omega$,

[TABLE]

It transpires that this possibility occurs only for $\omega\leq 4$ and when $a=1$. For $\omega\geq 5$, the optimum in each bin is given by

[TABLE]

Obviously $B_{2}(j)\leq B_{2}(K)\leq 0.391292208^{\frac{1}{2}}=0.625533539$, and we find

[TABLE]

Resulting bounds for $\sup_{j}\|\underline{x}^{\operatorname{fix}}\|_{\infty,\omega}$ are recorded in the following table

[TABLE]

We can thus update the bound for bias statistics $\beta_{k}(2)$ according to Lemma 6. Write, for $k=1,2,3,...$,

[TABLE]

for the local factor at $p$ that occurs at the $k$th bias statistic. Then the new bound becomes

[TABLE]

where $e_{j}$ indicates the $j$th elementary symmetric function. For large $j$ we use the bound $e_{j}(\underline{\tau})\leq\frac{e_{1}(\underline{\tau})^{j}}{j!}$. In this way we calculate that

[TABLE]

6. Asymptotic estimates

Recall that $P_{2}=4000$. For all $i\geq 2$ we let $P_{i+1}=P_{i}^{1.5}$. In this section we use the following explicit estimates for sums and products over primes, which hold for $i\geq 2$.

[TABLE]

These are verified in Appendix B.

For the remainder of the argument our inductive assumption is, for $i\geq 2$,

[TABLE]

Note that both of these hold at $i=2$.

Setting $M=2.949873427$ in Theorem 8, we estimate

[TABLE]

Appealing to Lemma 10 we bound the sums in $\mathscr{C}$ and $\mathbf{E}\mathscr{S}$, implicitly defined in (15), by

[TABLE]

[TABLE]

Combined with the asymptotics of $\mathbf{E}\|\underline{G}(\underline{0})\|_{\ell^{p}}^{p}$ from Lemma 9,

[TABLE]

we deduce

[TABLE]

Choose $C_{3}=\frac{0.7}{0.1023637064}$, $C_{2}=\frac{0.2}{0.6144485964}$, $C_{\operatorname{op}}=\frac{0.1}{0.0005048197920}$ so that the expectation is bounded by 1. As before, declare $r\in R_{i-1}^{*}\cap R_{i}$ to be good if

[TABLE]

Evidently $\pi_{\operatorname{good}}(i)\geq 0.3$, and for good $r$,

[TABLE]

but, again, we bin to get a stronger result.

For $K=100$ and integers $0<i,j$, $i+j\leq K+1$ let the bin $\mathscr{B}_{i,j}$ be those $r$ for which

[TABLE]

For $r\in\mathscr{B}_{i,j}$ we have

[TABLE]

We proceed much as before (again replacing $r$ with $i,j$ in each argument) updating bin-by-bin

[TABLE]

and

[TABLE]

We check bin-by-bin that

[TABLE]

so that our choice of $M=2.949873427$ in Theorem 8 is valid.

In each bin we solve the optimization problem (17) with $p=3$, and we find that for all $\omega\geq 1$ and for all bins the optimum is

[TABLE]

so that we guarantee

[TABLE]

We calculate

[TABLE]

Thus we find the following bounds.

[TABLE]

In Appendix B we verify that, for $i\geq 2$,

[TABLE]

Hence,

[TABLE]

and we find

[TABLE]

and

[TABLE]

On the other hand,

[TABLE]

and

[TABLE]

so that (18) is preserved, which completes the proof by induction.

Appendix A Symmetric functions

We briefly describe how we performed the calculations in the initial stage of the argument, see Section 5. There we appealed to Theorem 3, which is Theorem 2 with set $[n]$ identified with $\{p:4<p<222\}=p_{1}<p_{2}<...<p_{n}$ and weights $\pi_{p}=\frac{1}{p-1}$. We identify square-free number $m$ with the set of its prime factors. The Shearer functions

[TABLE]

are easily computed via

[TABLE]

with the $e_{i}$ elementary symmetric functions (take $e_{0}=1$), see [14].

The bias statistics are also not difficult to bound. Recall that $Q=\operatorname{LCM}(m:m\in\mathscr{M})$ and that $Q_{1}$ is the part of $Q$ composed of primes less than $P_{1}=222.$ The $k$th bias statistic is

[TABLE]

Let $\operatorname{sqf}(m)=\prod_{p\in S}p=:m_{S}$. Appealing to (4) of Theorem 3 we have

[TABLE]

so that

[TABLE]

Recall that we define

[TABLE]

Define for $i+j\leq n$ the mixed symmetric functions $f_{i,j}(\underline{\pi},\underline{\tau}_{k})$ by

[TABLE]

or, equivalently, by

[TABLE]

The sum of (19) is a linear combination of the mixed symmetric functions $f_{i,j}(\underline{\pi},\underline{\tau}_{k})$

[TABLE]

and so is rapidly computable.

Appendix B Explicit prime number estimates

In this appendix we sketch proofs for explicit bounds on well-known prime sums and products. Recall $P_{2}=4000$ and, for $i\geq 2$, $P_{i+1}=P_{i}^{1.5}$. In particular, no $P_{i}$ is prime. Let $\gamma$ denote the Euler-Mascheroni constant. Dusart [2], Theorem 6.12 proves the following estimate.

Theorem 11**.**

For $x>1$ we have

[TABLE]

and, for $x\geq 2973$ we have

[TABLE]

As a consequence, we obtain

Corollary 12**.**

For $i\geq 2$ we have

[TABLE]

For the sums of reciprocals of squares of primes, we have the following estimate.

Proposition 13**.**

For $i\geq 2$,

[TABLE]

and

[TABLE]

Proof.

We prove only the first inequality, as the second is similar. One easily checks

[TABLE]

For $i\geq 4$ use $\frac{0.99999997}{(p-1)^{2}}<\frac{1}{p^{2}}$ so that, for $x\geq P_{4}$,

[TABLE]

By [2], we have the inequality $|\theta(x)-x|<0.2\frac{x}{(\log x)^{2}}$ for $x\geq 3594641$. In particular, $0.99913x<\theta(x)<1.00088x$ in this range. Also, Lemma 9 of [11] yields

[TABLE]

Combined, these estimates give the claim. ∎

Recall that we define $\tau_{k}(x)=\sum_{i=1}^{\infty}\frac{(i+1)^{k}-i^{k}}{x^{i}}$. We have

[TABLE]

Proposition 14**.**

For $i\geq 2$,

[TABLE]

and

[TABLE]

Proof.

For $i=2,3$ this is verified directly. For $x\geq P_{4}$ this is a consequence of the following estimate of [2], Theorem 6.11. ∎

Theorem 15**.**

There is a constant $B$, such that, for $x>10372$,

[TABLE]

Appendix C Theorems of Lovász and Shearer-type

The Lovász local lemma considers the following scenario. In a probability space $\mathscr{X}$ there are events $\{A_{v}\}_{v\in V}$ with dependency graph $G=(V,E)$, that is, $A_{v}$ is independent of the $\sigma$-algebra $\sigma(A_{w}:(v,w)\not\in E)$. One seeks a positive lower bound for $\mathbf{P}(\bigcap_{v\in V}\overline{A_{v}})$. The local lemma guarantees that if there exist weights $1>x_{v}\geq\mathbf{P}(A_{v})$ satisfying

[TABLE]

then

[TABLE]

Shearer [13] gives an optimal bound of the above type via the independent set polynomial

[TABLE]

where $(v_{i}\sim v_{j})$ means $v_{i}\neq v_{j}$ and $(v_{i},v_{j})\not\in E$.

Theorem** (Shearer’s Theorem).**

Given $S\subset V$ let $\Xi_{S}(z_{v}:v\in V)$ denote $\Xi$ with arguments $z_{v}:v\in S$ replaced by 0. Subject to

[TABLE]

it holds

[TABLE]

Although Shearer’s Theorem is tight, evaluating the independent set polynomial is difficult and so there remains interest in finding statements of a similar type to the local lemma, which is more easily applied.

One way to reduce the complexity of Shearer’s theorem is to organize the events into collection of cliques. We consider the scenario in which graph $G=(V,E)$ is covered by a collection of cliques $\mathscr{K}$, that is, $E=\bigcup_{K\in\mathscr{K}}E_{K}$. For $v\in V$, let

[TABLE]

We make the assumption that $p(v)$ uniquely determines $v$ and we assume that all vertices have self-loops. What is the same, we take $V=\mathscr{P}(\mathscr{K})\setminus\{\emptyset\}$ to be the collection of all non-empty subsets of $\mathscr{K}$, and, for $S_{1},S_{2}\in V$, set $(S_{1},S_{2})\in E$ if and only if $S_{1}\cap S_{2}\neq\emptyset$. Consider vertex variables $(z_{v})_{v\in V}$ and clique variables $(\theta_{K})_{K\in\mathscr{K}}$. For $v\in V$, set also $\theta_{v}=\prod_{K:v\in K}\theta_{K}$. The clique partition function is defined to be

[TABLE]

Evidently $\Xi({\mathbf{v}},{\boldsymbol{\theta}})$ specializes to $\Xi({\mathbf{v}})$ at ${\boldsymbol{\theta}}=1$. Using this, we prove a clique version of Shearer’s theorem.

Theorem 16** (Clique Shearer Theorem).**

Let events $\{A_{v}:v\in V\}$ in probability space $\mathscr{X}$ have dependency graph $G=(V,E)$ covered by cliques $\mathscr{K}$ as above. For $S\subset\mathscr{K}$ define event $B_{S}=\bigcup_{v:p(v)\subset S}A_{v}$. Subject to the condition

[TABLE]

we have for all $\emptyset\subset S\subset T\subset\mathscr{K}$,

[TABLE]

Remark*.*

As compared to Shearer’s Theorem, the clique Shearer Theorem has the advantage that the number of conditions which must be checked is exponential in the number of cliques, rather than in the number of vertices.

Proof.

The proof is by induction. Let $S\subset\mathscr{K}$ and suppose the conclusion holds for subsets $T\subset S$. Let $K\in\mathscr{K}\setminus S$. Then

[TABLE]

∎

As a consequence we obtain a proof of a generalization of Theorem 3.

Theorem** (Shearer-type theorem).**

Suppose we have a probability space and a parameter $\theta\geq 1$. Let $[n]=\{1,2,...,n\}$, and assume that for each $1\leq i\leq n$ there is a weight $\pi_{i}$ assigned, satisfying $\frac{1}{\theta}\geq\pi_{1}\geq\pi_{2}\geq...\geq\pi_{n}\geq 0$. Let the sets $\emptyset\neq T\subset[n]$ index events $A_{T}$ each having probability

[TABLE]

Assume that $A_{T}$ is independent of $\sigma(\{A_{S}:S\subset[n],S\cap T=\emptyset\})$, so that a valid dependency graph for the events $\{A_{T}:\emptyset\neq T\subset[n]\}$ has an edge between $S\neq T$ whenever $S\cap T\neq\emptyset$.

Define $\rho_{\theta}(\emptyset)=1$, and given $\emptyset\neq T\subset[n]$, set

[TABLE]

Suppose that $\rho_{\theta}([1])\geq\rho_{\theta}([2])\geq...\geq\rho_{\theta}([n])>0.$ Then for any $\emptyset\neq T\subset[n]$,

[TABLE]

and, for any $T_{1}\subset T_{2}\subset[n]$,

[TABLE]

Proof.

It is observed in [14] that $\rho_{\theta}(T)$ may be expressed as a linear combination of elementary symmetric functions in $\{\pi_{t}:t\in T\}$. Indeed, if $B(m,j)$ denotes the generalized Bell number, that is, the number of ways of partitioning a set of size $m$ into $j$ parts then, setting $|T|=M$ and making the convention $e_{0}(\underline{\pi})=1$,

[TABLE]

where $X_{\theta}$ satisfies the recurrence

[TABLE]

In particular, as exploited in [12], $\rho(T)$ is affine linear in each variable $\pi_{t}$.

We check that under the given conditions, $\rho_{\theta}(T)>0$ for any $T\subset[n]$, which reduces this theorem to the clique Shearer Theorem.

Given vectors $\underline{x},\underline{y}\in\mathbb{R}^{m}$, say that $\underline{x}\leq\underline{y}$ if $x_{i}\leq y_{i}$ for each $i$. By induction, we show that for any $1\leq m\leq n$ and for $\underline{0}\leq\underline{x}\leq\underline{\pi}$, $\rho_{\theta}(\underline{x})\geq\rho_{\theta}(\underline{\pi})>0$, from which the case for $T$ follows since the $\pi_{i}$ are decreasing.

When $m=1$, $\rho_{\theta}(\pi_{1})=1-\theta\pi_{1}\leq 1-\theta x_{1}=\rho_{\theta}(x_{1})$. Given $m>1$, assume inductively the statement for all $m^{\prime}<m$.

Note that, by hypothesis, we have $\rho_{\theta}(\pi_{1},...,\pi_{m-1})\geq\rho_{\theta}(\pi_{1},...,\pi_{m})>0$.

We show by an inner induction that for $1\leq j\leq m$,

[TABLE]

When $j=1$ this holds, since $(\pi_{2},...,\pi_{m})\leq(\pi_{1},...,\pi_{m-1})$ so that, by the inductive assumption

[TABLE]

from which

[TABLE]

follows by affine linearity.

Having shown

[TABLE]

the case

[TABLE]

again follows by affine linearity from

[TABLE]

∎