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IRREDUCIBILITY OVER THE MAX-MIN SEMIRING
BENJAMIN BAILY, JUSTINE DELL, HENRY L. FLEISCHMANN, FAYE JACKSON, STEVEN J. MILLER,
ETHAN PESIKOFF, AND LUKE REIFENBERG
ABS TR ACT. For sets A, B ⊂N, their sumset is A+B:= {a+b:a∈A, b ∈B}. If we cannot write a set
Cas C=A+Bwith |A|,|B|≥ 2, then we say that Cis irreducible. The question of whether a given set
Cis irreducible arises naturally in additive combinatorics. Equivalently, we can formulate this question as one
about the irreducibility of boolean polynomials, which has been discussed in previous work by K. H. Kim and
F. W. Roush (2005) and Y. Shitov (2014). We prove results about the irreducibility of polynomials and power
series over the max-min semiring, a natural generalization of the boolean polynomials.
We use combinatorial and probabilistic methods to prove that almost all polynomials are irreducible over the
max-min semiring, generalizing work of Y. Shitov (2014) and proving a 2011 conjecture by D. Applegate, M.
Le Brun, and N. Sloane. Furthermore, we use measure-theoretic methods and apply Borel’s result on normal
numbers to prove that almost all power series are asymptotically irreducible over the max-min semiring. This
result generalizes work of E. Wirsing (1953).
CONTENTS
1. Introduction 1
2. Proof of Theorem 1.8 2
3. Proof of Theorem 1.9 7
References 10
1. INTRODUCTION
The max-min semiring is defined as N= (N∪{∞},⊕,⊗), where a⊕b= max(a, b), a ⊗b= min(a, b).
Previous work ([KR05], [Shi14]) has discussed the factorization of polynomials over the Boolean Semiring,
that is, the subsemiring B2={0,1}. In this restricted case, the two questions we hope to settle in general
have already been resolved.
Definition 1.1. Let f∈ N[[x]]. If f=gh implies that either gor his a monomial, then fis irreducible.
Theorem 1.2 (Shitov, 2014).As n→ ∞, the proportion of degree npolynomials in B2[x]which are
irreducible tends to 1.
This result answers a 2005 question of K. H. Kim and F. W. Roush. Remarkably, the proof uses only
elementary combinatorics and probability.
Definition 1.3. Let f, g ∈ N[[x]]. If fand gdiffer in only finitely many coefficients, then we say f∼g.
Definition 1.4. Let f∈ N [[x]]. If f∼gh implies that either gor his a monomial, then fis asymptotically
irreducible.
Theorem 1.5 (Wirsing, 1953).Almost every element of B2[[x]] is asymptotically irreducible.
Date: November 19, 2021.
2020 Mathematics Subject Classification. 11B30, 11R09, 15A80.
This work was supported by NSF Grants DMS1561945 and DMS1659037. We thank the participants of the 2021 Williams
SMALL REU for constructive comments. Thanks also to Leo Goldmakher for translating Wirsing’s paper from the original German
and for helpful feedback throughout.
1
arXiv:2111.09786v1 [math.CO] 18 Nov 2021
2 B. BAILY, J. DELL, H. L. FLEISCHMANN, F. JACKSON, S. J. MILLER, E. PESIKOFF, AND L. REIFENBERG
This proof is measure-theoretic and builds heavily off the work of Borel, in particular the result that almost
every number is normal (definition 3.5) in base 2. Interestingly, Wirsing and Shitov phrased these results in
two different settings. Wirsing in fact writes that almost every set A⊂Nis asymptotically irreducible.
Definition 1.6. Let A, B ⊂Gfor some group G. Then A+B={a+b:a∈A, b ∈B}.
Definition 1.7. Let S⊂N. If S=A+Bimplies either Aor Bis a singleton, then Sis irreducible.
Similarly, if S∼A+Bimplies Aor Bis a monomial, then Sis asymptotically irreducible. Here, ∼
denotes difference in finitely many elements just like before.
We restate Shitov and Wirsing’s results in these terms: almost every finite subset of Nis irreducible, and
almost every subset of Nis asymptotically irreducible. The semiring of sets under union and set addition is
isomorphic to the semiring of boolean polynomials [Gro19], hence these two formulations are equivalent.
Our contribution is to generalize these results to the wider setting of the max-min semiring.
Theorem 1.8. Fix b, and set Bb={0,1, . . . , b −1} ⊂ N. Then as n→ ∞, the proportion of degree n
polynomials in Bb[x]which are irreducible tends to 1.
Theorem 1.9. Almost every element of Bb[[x]] is asymptotically irreducible.
Just as products of boolean polynomials correspond to sums of sets, products of min-max polynomials
correspond to sums of multisets. For more details on this correspondence, see [Gro19].
In Section 2, we prove Theorem 1.8 by partitioning the collection of reducible polynomials in Bb[[x]] into
several subcollections and bounding the size of each. We do this by applying Hoeffding’s inequality and
a generalization of a lemma from [Shi14]. In Section 3, we prove Theorem 1.9 by partitioning the set of
reducible elements of Bb[[x]] into subcollections and bounding the size of each. This is done via applications
of the Borel-Cantelli Lemma and the result that almost all numbers are normal in every base [Wei21].
2. PRO OF OF TH EOREM 1.8
In this section, we generalize Shitov’s result to polynomials over the max-min semiring. We begin with
some conventions.
Definition 2.1. Let f∈ N[x]. Then |f|denotes the number of nonzero coefficients of f.
Definition 2.2 ([ALS11]).Adigit map is a nondecreasing function N→N. If dis a digit map and f=
a0⊕a1x⊕a2x2⊕ ··· ∈ N[[x]], then we let d(f) = d(a0)⊕d(a1)x⊕d(a2)x2⊕ · ··.
Proposition 2.3 ([ALS11]).If dis a digit map, then d:N[[x]] → N[[x]] is a semiring homomorphism.
In particular, if f=gh is a nontrivial factorization and d(1) ≥1, then d(f) = d(g)d(h)is a nontrivial
factorization of d(f).
This key idea provides a powerful framework for our proof, allowing us to partially reduce the problem
of factoring a polynomial over Bbto factoring one over B2.
Definition 2.4. We define the digit maps maps sifor each i∈Z+.
si(n) := (0n<i
i n ≥i. (2.1)
For f∈ N[[x]], we additionally define fi=s1(si(f)). These polynomials, which we refer to as the “i-level
support of f”, are indicator functions for where the coefficients of fare at least i.
Finally, to conclude our setup, we use the following convention for referencing the coefficients of poly-
nomials.
IRREDUCIBILITY OVER THE MAX-MIN SEMIRING 3
Definition 2.5. Throughout the remainder of this paper, let
f=
∞
M
k=0
αkxk, g =
∞
M
k=0
βkxk, h =
∞
M
k=0
γkxk, σ =
∞
M
k=0
δkxk.
Additionally, set α0
i=αi⊗1and similarly for each other coefficient. This way we have, for instance:
f1=
∞
M
k=0
α0
kxk, g1=
∞
M
k=0
β0
kxk, h1=
∞
M
k=0
γ0
kxk, σ1=
∞
M
k=0
δ0
kxk.
In the proof of another lemma, Shitov shows the following statement, which will be of great use to us.
Corollary 2.6 ([Shi14]).For any d > 0, the number of pairs of boolean polynomials (f , g)satisfying the
following conditions is at most n2d+12(k,n).
(1) The constant terms of f, g are nonzero;
(2) deg f=k > 0,deg g=n−k;
(3) |f⊗g|≤|f|+|g|+d.
Due to our slightly different convention about irreducibility, we require a version of this lemma when the
condition (1) is not necessarily satisfied. We now address the opposing conventions of irreducibility.
Our definition of irreducible is slightly broader than the traditional notion of irreducibility over a semiring
(f=gh =⇒gis a unit). The advantages of our definition are as follows.
•If f∈ Bb[[x]] is irreducible, then fis also irreducible over N[[x]]. In contrast, f= (b−1) ⊗f
is trivial factorization over Bb[[x]], but a nontrivial factorization over N[[x]] as b−1is no longer a
unit. Thus, despite introducing no new factors, fis now reducible.
•The Taylor series fsuch that g7→ f⊗gis an injective endomorphism of any semiring Bb[[x]]
are precisely the monomials. Thus, despite not having multiplicative inverses, multiplication by
monomials is invertible in the sense that we have cancellation.
•This definition lines up with the additive combinatorics. For instance, the set {1,2,4}is additively
indecomposible, but the polynomial x⊕x2⊕x4is reducible as x(1 ⊕x⊕x3)under previous
authors’ definitions.
We now generalize Corollary 2.6 to our setting.
Lemma 2.7. The number of pairs boolean polynomials (f, g)satisfying the following conditions is at most
n2d+22kfor any d > 0.
(1) The constant term of fis nonzero;
(2) deg f=k > 0,deg g=n−k;
(3) |f⊗g|≤|f|+|g|+d.
Proof. Write g=xj⊗(1 + ·· · +xn−k−j)and define gby g=xj⊗g. Then clearly |f⊗g|=|f⊗g|
and |g|=|g|. By Corollary 2.6, there are at most n2d+12(k,n−j)pairs (f, g)satisfying the hypotheses of the
corollary. Since there are at most nchoices for j, the number of pairs (f, g )satisfying the hypotheses of this
lemma is at most Pn−1
j=0 n2d+12(k,n−j)≤n2d+2 2k.
The final ingredient for our proof is Hoeffding’s inequality: a probabilistic lemma which Shitov used, in
conjunction with Corollary 2.6, to prove Theorem 1.2.
Proposition 2.8 (Hoeffding’s Inequality).Let Xnbe a sum of nindependent Bernoulli random variables
Xwith E[X] = p. Then P(|Xn−np|> n)≤2e−22n.
Proof. See [Hoe63], Theorem 2.
4 B. BAILY, J. DELL, H. L. FLEISCHMANN, F. JACKSON, S. J. MILLER, E. PESIKOFF, AND L. REIFENBERG
When we choose a degree n−1polynomial fat random from Bb[x], the quantity |fi|is a sum of n
independent Bernoulli random variables Ziwith E[Zi] = b−i
b. As a consequence, if fis a degree n−1
polynomial chosen randomly from Bb[x], then
P|fi| − (b−i)n
b> n≤2e−22n.(2.2)
Definition 2.9. If f , g are nonnegative real-valued functions and there exists a constant c > 0such that
f≤cg, then we write f.g.
We now prove a quantitative version of Theorem 1.8.
Proposition 2.10. Let b > 1and a=bb/2cand let Σb,n denote the set of reducible degree n−1polynomials
in Bb[x]. Then for any d, v > 0we have
|Σb,n|.bnne−d2/4(n+1) +vn2d+12vb−n+n22−v+n2d+3 2d
2−n
3.
Proof. We will partition Σb,n into 7 sets Σb,n ⊂E1
n(d, v)∪ · ·· ∪ E7
n(d, v). Our proposition follows from
the bound |Σb,n| ≤ E1
n(d, v)+·· · +E7
n(d, v).
We now detail the partition. Though we will not write this after each set, we stipulate that h∈Ei
n(d, v)
only if h /∈Ej
n(d, v)for any j < i.
E1
n(d, v)is the set of polynomials hsuch that |h1| − (b−1)n
b>d
2.
E2
n(d, v)is those h=f⊗gsuch that |f1|+|g1| − (b−1)(n+1)
b>d
2.
E3
n(d, v)is those hsuch that |ha| − (b−a)n
b>d
2.
E4
n(d, v)is those h=f⊗gsuch that |fa|+|ga| − (b−a)(n+1)
b>d
2.
The size of each of these sets can be bounded using Equation (2.2). We start by considering these
sets in order to control the size of the supports of the polynomials in the remaining sets. In par-
ticular, we want |hi|≤ |fi|+|gi|+dfor i= 1, a. This is so that when hi=fi⊗giis a nontrivial
factorization, the hypotheses of Lemma 2.7 apply and we can conclude that there are few possible
pairs (f, g).
E5
n(d, v)is those h=f⊗gwith deg f≤v.
E6
n(d, v)is those h=f⊗gwith |fa| ≤ 1or |ga| ≤ 1.
The set E6
n(d, v)contains polynomials h=f⊗gwhere ha=fa⊗gais a trivial factorization.
However, if deg f≤v, then h∈E5
n(d, v), and thus not in E6
n(d, v). Using this fact allows us to
achieve the necessary upper bound on the size of E6
n(d, v).
E7
n(d, v)is all remaining reducible degree n−1polynomials h. Once the first 6 sets are considered,
every remaining reducible polynomial h=f⊗gsatisfies |ha|≤ |fa|+|ga|+d, and ha=fa⊗ga
is a nontrivial factorization. Thus, we are able to apply Lemma 2.7 to conclude that the number of
remaining reducible polynomials is small.
Now, we are ready to bound the size of each of these sets.
(1) By Equation (2.2) with =d
2n, we obtain
E1
n(d, v),E3
n(d, v)≤2e−d2/4nbn.e−d2/4nbn≤e−d2/4(n+1)bn.
(2) Each pair (f, g )corresponds to only one choice of h, thus it suffices to bound the number of pairs
(f, g). If we fix deg f=k, then we must have deg g=n−k−1as deg(f⊗g) = n−1. The
IRREDUCIBILITY OVER THE MAX-MIN SEMIRING 5
set of pairs (f, g)∈(Bb[x])2such that deg f=k, deg g=n−k−1is in bijection with the set of
degree npolynomials of Bb[x], with the bijection given below:
φ(f, g) = f⊕((b−1)xk+1)⊗g
φ−1(h) =
k
M
j=0
γjxj,
n
M
j=k+1
γjxj−k−1
.
Moreover, |fi|+|gi|=|(φ(f, g))i|. Thus, choosing =d
2n+2 and applying Equation (2.2), we
obtain that there are at most 2e−d2/4(n+1)bn+1 such pairs (f , g). Since there are n/2choices for
deg f, we use this bound for each choice and obtain
E2
n(d, v),E4
n(d, v)≤ne−d2/4(n+1)bn+1 .ne−d2/4(n+1)bn.
(3) Let h=f⊗g∈E5
n(d, v). Since h /∈E1
n(d, v)∪E3
n(d, v), we have |h1| ≤ (b−1)n
b+d
2,|f1|+|g1| ≥
(b−1)(n+1)
b−d
2. Thus |h1|≤|f1|+|g1|+d. If |f1| ≤ 1or |g1| ≤ 1, then for gis a monomial
in contradiction to the assumption that f⊗gis a nontrivial factorization, hence (f1, g1)satisfy
every hypothesis of Lemma 2.7. We apply this lemma once for each choice of 1≤deg f≤v, and
conclude
E5
n(d, v)≤
v
X
deg f=1
n2d+12deg f≤vn2d+12v.
(4) Suppose |fa| ≤ 1. Then fix deg f=k. Since h /∈E5
n(d, v), we can assume k > v. Then there are
(k+1)(b−a)(a−1)kchoices1for fand bn−kchoices for g, hence there are ≤(k+1)(a−1)kbn−k+1
pairs (f, g). There are at most nchoices for k, hence
E6
n(d, v)≤
n
X
k=v
(k+ 1)(a−1)kbn−k+1 ≤n(n+ 1)(a−1)vbn−v+1
.n2(a−1)vbn−v≤n2b
2v
bn−v≤n22−vbn.
If instead |ga| ≤ 1, then we have that deg(g)≥deg(f)≥v. By symmetry, there are at most twice
as many pairs with either |fa|≤ 1or |ga|≤ 1as there are with |fa|≤ 1. This doubles the size of our
upper bound, but this is only a constant factor.
(5) Let h=f⊗g. Since h /∈E3
n(d, v)∪E4
n(d, v), we have |ha|<(b−a)n
b+d
2and |fa|+|ga|>
(b−a)(n+1)
b, hence |ha| ≤ |fa|+|ga|+d. Moreover, as h /∈E6
n(d, v), neither fanor gais a
monomial and thus the pair (fa, ga)is a nontrivial factorization of haand satisfies the hypotheses
of Lemma 2.7. Thus, using the fact that (deg fa, n −1) ≤n−1
2≤n
2for 1≤deg f≤n−2, the
number of possible choices for hais at most
n−2
X
deg fa=1
n2d+22(deg fa,n−1) ≤n2d+32n
2.
Once hais known, if |ha|=k, there are an−k(b−a)kchoices for h. This is because each 0
coefficient of hacan correspond to any coefficient in {0, . . . , a −1}, and any 1 corresponds to a
coefficient in {a, . . . , b −1}. Since h /∈E3
n(d, v), we can say k≤(b−a)n
b+d
2, a quantity which we
1Pick the index of the coefficient to be at least a, then pick its value, then pick the remaining coefficients from {0,· · · , a −1}.
6 B. BAILY, J. DELL, H. L. FLEISCHMANN, F. JACKSON, S. J. MILLER, E. PESIKOFF, AND L. REIFENBERG
denote by sto clean up our expressions. Recalling that a:= bb/2c, we have a≤b/2≤(b−a),
with equality of all terms when bis even. This gives the following upper bound:
an−k(b−a)k≤an−s(b−a)s
≤aan
b(b−a)(b−a)n
bb−a
ad
2
≤an
2(b−a)n
2b−a
a(b−a
b−1
2)nb−a
ad
2
≤b
2nb−a
a(b−a
b−1
2)nb−a
ad
2.
For b≥2, we have the bounds 1≤b−a
a≤2and 0≤b−a
b−1
2≤1
6, both of which are achieved
when b= 3. Moreover, for b= 2, we have b−a
b
b−a
b−1
2= 1, thus for any b≥2we have
b−a
b(b−a
b−1
2)n≤2n
6. Altogether, this yields:
E7
n(d, v)≤n2d+32n
2b
2nb−a
ad
2b−a
b(b−a
b−1
2)n
≤n2d+32n
2b
2n
2n
62d
2≤n2d+32d
2−n
3bn.
With the right choice of d, v, this gives us a proof of Theorem 1.8.
Proof. Our goal is to show that |Σb,n |
bn→0, from which the result follows. Set d= 2√n+ 1 log nand
v= 3 log2nthen apply Proposition 2.10. We show that each summand of the upper bound on |Σb,n|
bn
vanishes.
(1) We have ne−d2/4(n+1) =ne−(log n)2=n1−log n, which vanishes as n→ ∞.
(2) We have log(vn2d+12vb−n) = log v+ (4√n+ 1 log n+ 2) log n+ 3 log2nlog 2 −nlog b. Each
summand of this expression is sub-linear except for the one which is negative, therefore this diverges
to −∞. It follows that
(vn2d+12vb−n) = elog(vn2d+1 2vb−n)→0.
(3) We have n22−v=n22−3 log2n=n−1→0.
(4) We have log n2d+32d
2−n
3= 4√n+ 1 log n+ (√n+ 1 log n−n
3) log 2. By the same reasoning
as the bound on the second summand, we conclude n2d+32d
2−n
3→0.
This result in hand, we are now prepared to state and prove the conjecture of Applegate, LeBrun, and
Sloane [ALS11]. Their conjecture refers to prime elements of the semiring, which we define below, and is
in some ways a more natural definition.
Definition 2.11. A polynomial h∈ Bb[x]is prime if h=f⊗gimplies either f, g =b−1.
Conjecture 2.12. Let πb(n)denote the number of degree n−1prime polynomials of Bb[x]. Then πb(n)∼
(b−1)2bn−2.
IRREDUCIBILITY OVER THE MAX-MIN SEMIRING 7
Motivating their conjecture, Applegate et al. observed that only certain polynomials can be prime.
Definition 2.13. Aprime candidate of Bb[x]is a polynomial with nonzero constant term and maximum
coefficient b−1.
It is easy enough to see that a polynomial is prime only if it is a prime candidate. If h=ajxj⊕ · ·· ⊕
an−1xn−1for j > 1, then h= (b−1)xj⊗(aj⊕ · · · ⊕ an−1xn−j−1)which is a nontrivial factorization in
their convention. Moreover, if c < b −1is the maximum coefficient of h, then h=c⊗h.
They showed that the number of prime candidates is asymptotic to (b−1)2bn−2, and from their data2, as
k→ ∞, almost all prime candidates are in fact prime. As evidence for this fact, Applegate et al. produced
the following lower bound:
(b−1)n−2+ 2(b−2)n−2+··· ≤ πb(n).
Moreover, they observed the following, which we will re-prove here.
Lemma 2.14 ([ALS11]).An irreducible prime candidate is prime.
Proof. If his irreducible, then h=fg implies either f , g is a monomial, without loss of generality, fis.
Since the constant term of his nonzero, we must have that fis a constant. Since the maximum coefficient
of his b−1, we must also have that f=b−1, thus his prime.
With this lemma, Conjecture 2.12 is a simple corollary of Theorem 1.8
Proof. The proportion of degree n−1prime candidates of Bb[x]which are irreducible is at most a quantity
which vanishes as n→ ∞:|Σb,n|
(b−1)2bn−2.|Σb,n|
bn→0.(2.3)
It follows that almost all prime candidates are prime.
3. PRO OF OF TH EOREM 1.9
Before we prove this, we first must clarify what we mean by “almost all.” It turns out, there is a very
natural measure to associate to the set Bb[[x]].
Definition 3.1. To each element of Bb[[x]] we associate a real number in [0, b], given by
ρb ∞
M
k=0
akxk!:=
∞
X
k=0
akb−n.(3.1)
In other words, each power series corresponds to a string of digits in [0,1, . . . , b −1], which we can
interpret as the base-bexpansion of a number. This allows us to define a probability measure mon Bb[[x]].
Definition 3.2. For a set A⊂ Bb[[x]] such that ρb(A)is a measurable subset of R, let m(A) = b−1L(ρb(A)),
where Ldenotes the Lebesgue measure.
This reframing allows us to ask and answer questions about these polynomials measure-theoretically. For
example, we will use Borel’s theorem that every number is normal, regardless of base [Wei21].
We deduce Theorem 1.9 from a second theorem.
Theorem 3.3. Let Cb⊂ Bb[[x]] denote the set of reducible polynomials. Then m(Cb)=0.
We show first how Theorem 1.9 follows from Theorem 3.3.
Proof. For f∈ Bb[[x]], let [f]denote the set of all gsuch that f∼g. The set of asymptotically reducible
fis precisely the set [Cb]. Fix a natural number n, and notice the set of functions which differ in exactly
ncoefficients from some element of Cbhas measure 0. Thus, [Cb]is a countable union of measure 0 sets,
hence it has measure 0 and almost all power series over Bb[[x]] are irreducible.
2OEIS sequences (A169912),(A087636) show the number of prime elements of B2[x],B10[x]of each degree n.
8 B. BAILY, J. DELL, H. L. FLEISCHMANN, F. JACKSON, S. J. MILLER, E. PESIKOFF, AND L. REIFENBERG
We now prove Theorem 3.3. Our proof parallels Wirsing’s original argument to a great extent, but as the
authors are not aware of an English translation of Wirsing’s result [Wir53], we reproduce it here for the sake
of completeness.
Definition 3.4. For n∈Nand f=L∞
k=0 akxk∈ N[[x]], define f(n) := Ln
k=0 akxk∈ N[x].
First, partition Cbinto three sets Tb
1, T b
2, T b
3:
Tb
1:= {h:h=f⊗gwith 2≤ |g1|<∞}
Tb
2:= h:h=f⊗gwith lim inf
n→∞ |f1(n)|+|g1(n)|
n<1
5and |f1|=∞=|g1|
Tb
3:= h:h=f⊗gwith lim inf
n→∞ |f1(n)|+|g1(n)|
n≥1
5and |f1|=∞=|g1|.
Since Tb
1∪Tb
2∪Tb
2=Cb, it suffices to show that L(Tb
1) = L(Tb
2) = L(Tb
3) = 0. In proving that the
measures of Tb
1and Tb
3are 0, we rely extensively on the following idea.
Definition 3.5. A number λ∈Ris normal in base bif the base brepresentation of λcontains an equal
proportion of each finite sequence of digits base b. That is, if for all positive integers n, all possible strings
of ndigits have density b−nin the base brepresentation.
More formally, let s= (δ1, . . . , δk)be a string of digits in {0, . . . , b −1}. Fix a real number λand let
Nλ(n, s)denote the number of occurences of the string sin the first ndigits of the base-bexpansion of λ.
Then the following holds:
lim
n→∞
Nλ(n, s)
n=b−k.
An equivalent formulation of this is the following: let Z⊂ {0, . . . , b−1}kand let Nλ(n, Z) = Ps∈ZN(n, s).
Then
lim
n→∞
N(n, Z)
n=|Z|b−k.(3.2)
Theorem 3.6. (Borel, 1909 for base 2; Wirsing, 1953 for the general case) For any b≥2, almost every
λ∈Ris normal base b. Consequently, almost every λ∈Ris absolutely normal, that is, normal in every
base.
We now state an important lemma with an elementary proof.
Lemma 3.7. If h=f⊗g, then Ln
k=0 αk⊗βn−k=γn.
Proof. To elucidate this fact, all we need to do is rewrite the product f⊗g:
f⊗g=
∞
M
i=0
αixi
∞
M
j=0
βjxj=
∞
M
n=0 n
M
k=0
αk⊗βn−k!xn=
∞
M
n=0
γnxn.
Lemma 3.8. We have m(Tb
1) = 0.
Proof. We show that no element of Tb
1is normal, whence the result follows. Specifically, we claim that the
following sequence of digits can never occur in ρb(h)for any h=f⊗g∈Tb
1:
00 . . . 0
| {z }
deg g+1
1 00 . . . 0
| {z }
deg g+1
.(3.3)
Let f1=L∞
k=0 αkxk, g1=Ldeg g1
k=0 βkxk, h1=L∞
k=0 γkxk. We can write
h1=g1⊗f1=
deg g
M
k=0
βkxk⊗f1.
IRREDUCIBILITY OVER THE MAX-MIN SEMIRING 9
If γk= 1, then by Lemma 3.7 there exist i, j such that αi=βj= 1 and i+j=k. Since g1is not
a monomial, there exists another index j06=jsuch that βj0= 1. Then by Lemma 3.7:1≤γi+j0and
γ0
i+j0= 1. The gap between the two indices i+j, i +j0is at most deg g1(but either index can come
first), thus ρb(h1)does not have a “1” without another “1” at most deg gindices away. Thus the string
Equation (3.3) does not occur in ρb(h1).
Lemma 3.9. We have m(Tb
2) = 0.
Proof. We begin by defining a finite counterpart to Tb
2:
Tb
2(n) := ρb(h) : h=f⊗g:|f1(n)|+|g1(n)|
n<1
5and |f1|=∞=|g1|.
Notice that
Tb
2⊆lim sup({Tb
2(n)}) = \
N≥1[
n≥N
Tb
2(n).
By the Borel-Cantelli Lemma, we know that if
∞
X
n=1
m(Tb
2(n)) <∞,
then
mlim sup
n→∞ (Tb
2(n))=m(Tb
2)=0.
As such, it suffices to show that P∞
n=1 m(Tb
2(n)) <∞.
Fix an integer kand consider all possible fand hsuch that |f1(n)|+|g1(n)|=k. There are 2n+2
k
possibilities for f1(n)and g1(n): each has n+1 coefficients, and we distribute knonzero coefficients among
them. Additionally, for a given choice of f1(n)and g1(n), there are (b−1)kpolynomials f(n)and g(n)
since each 1 coefficient of f1or g1can correspond to any value in {1, . . . , b −1}. Thus, for a given k, there
are at most (b−1)k2n+2
kpossibilities for f(n)⊗g(n). Therefore, Tb
2(n)is a subset of a union of at most
P0≤k≤n
5(b−1)k2n+2
kintervals, each of length b−n.
We then compute
m(Tb
2(n)) ≤1
bnX
0≤k≤n
5
(b−1)k2n+ 2
k
≤n
5bn(b−1)n/52n+ 2
bn/5c
≤n
bn(b−1)n/52n
bn/5c
≤n
bn(b−1)n/52ne
n/5n/5
≤n
bn(10e(b−1))n/5
≤n 1.94(b−1)1/5
b!n
.
Notice that 1.94(b−1)1/5
b<1for b≥2. Hence, the sum P∞
n=1 m(Tb
2(n)) converges, so m(Tb
2)=0.
Lemma 3.10. We have m(Tb
3)=0.
10 B. BAILY, J. DELL, H. L. FLEISCHMANN, F. JACKSON, S. J. MILLER, E. PESIKOFF, AND L. REIFENBERG
Proof. As in the case of Tb
1, we will show that no element of Tb
3is normal, from which the result will follow.
Without loss of generality, we know that lim inf n→∞ |f1(n)|
n≥1
10 . Let kbe a positive integer such that
b−1
bk
<1
10.
Pick a positive integer rsuch that |g1(r−1)|=k. This is equivalent to choosing rsuch that ρb(g1(r−1))
has exactly kones. Let Zdenote the set of degree r−1polynomials in σ∈ Bb[x]such that σ1⊕g1=σ1.
In other words, Zis the set of degree r−1polynomials of σ∈ Bb[x]such that βi6= 0 =⇒δi6= 0. We can
compute |Z|using a counting argument: If βi6= 0, then δi∈ {1, . . . , b −1}, otherwise δi∈ {0, . . . , b −1}.
As |g1|=kand σhas rcoefficients, there are (b−1)kbr−kpossible choices for σ.
If ρb(h)is normal, we expect the digit strings in ρb(Z)to occur at a frequency of (b−1)kbr−k
br=b−1
bkin
ρb(h). We show that they instead occur at a frequency of at least 1
10 , from which it follows that ρb(h)is not
normal.
Suppose α0
s= 1. Then from Lemma 3.7, it follows that:
γ0
sxs⊕ · ·· ⊕ γ0
s+r−1xs+r−1⊕α0
sxs⊗g1(r−1)=α0
sxs⊗g1(r−1).
Thus γs⊕···⊕ γs+r−1xr−1∈Z. This observation allows us to lower-bound the frequency of these strings
in ρb(h):
1
10 ≤lim inf
n→∞ |f1(n)|
n≤lim inf
n→∞ Nρ(h)(n+r−1, Z)
n= lim inf
n→∞ Nρ(h)(n, Z)
n.
The above contradicts Equation (3.2), thus his not normal.
We now prove Theorem 3.3, from which Theorem 1.9 is a corollary.
Proof. By construction, Cb=Tb
1∪Tb
2∪Tb
3. As a consequence of Lemma 3.8, Lemma 3.9, and Lemma 3.10,
we have
m(Cb)≤m(Tb
1) + m(Tb
2) + m(Tb
3) = 0.
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REFERENCES 11
Email address:bmb2@williams.edu
DEPARTMENT OF MATHEMATICS AND STATISTICS, WILLIAMS CO LL EG E, WIL LI AM ST OWN , MA 01267
Email address:jdell@haverford.edu
DEPARTMENT OF MATHEMATICS AND STATISTICS, HAVE RF OR D COL LE GE , HAVERFORD, PA 19041
Email address:henryfl@umich.edu
DEPARTMENT OF MATHEM ATICS , UNIVERSITY OF MICHIGAN, ANN AR BO R, MI 48109
Email address:alephnil@umich.edu
DEPARTMENT OF MATHEM ATICS , UNIVERSITY OF MICHIGAN, ANN AR BO R, MI 48109
Email address:sjm1@williams.edu,Steven.Miller.MC.96@aya.yale.edu
DEPARTMENT OF MATHEM ATICS A ND STATISTICS, WILLIAMS COL LE GE , WI LL IA MS TOW N, MA 01267
Email address:ethan.pesikoff@yale.edu
DEPARTMENT OF MATHEM ATICS , YALE UNIVERSITY, NE W HAVEN , CT 06520
Email address:lreifenb@nd.edu
DEPARTMENT OF MATHEM ATICS , UNIVERSITY OF NOT RE DAME, NOTR E DAME, IN 46556