Topics in Multiplicative and Probabilistic Number Theory by Alexander P. Mangerel A thesis ...

285 Pages·2017·1.66 MB·English

Topics in Multiplicative and Probabilistic Number Theory
by
Alexander P. Mangerel
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Mathematics
University of Toronto
(cid:13)c Copyright 2018 by Alexander P. Mangerel
Abstract
Topics in Multiplicative and Probabilistic Number Theory
Alexander P. Mangerel
Doctor of Philosophy
Graduate Department of Mathematics
University of Toronto
2018
A heuristic in analytic number theory stipulates that sets of positive integers cannot simultaneously be
additively and multiplicatively structured. The practical verification of this heuristic is the source of a
great number of difficult problems. An example of this is the well-known Hardy-Littlewood tuples con-
jecture, which asserts that, infinitely often, one should be able to find additive patterns of fairly general
shapeintheprimes. Conjecturesofthistypearealsoatleastmorallyequivalenttotheexpectationthat
a multiplicative function, unless it has a special form, behaves randomly on additively structured sets.
In this thesis, we consider several problems involving the behaviour of multiplicative functions interact-
ing with additively structured sets. Two main topics are studied: i) the estimation of mean values of
multiplicative functions, i.e., the limiting average behaviour of partial sums of multiplicative functions
along an interval whose length tends to infinity; and ii) the estimation of correlations of multiplicative
functions, i.e., the behaviour of simultaneous values of multiplicative functions at arguments that are
additively related. A number of applications of the study of these topics are also addressed.
First, we prove quantitative versions of mean value theorems due to Wirsing and Hal´asz for multiplica-
tive functions that often take values outside of the unit disc. This has a broad realm of applications. In
particular we are able to extend a further theorem of Hala´sz, proving local limit theorems for vectors
of certain types of additive functions. We thus confirm a probabilistic heuristic in the small deviation
regime and beyond for the functions in question.
In a different direction, we consider the collection of periodic, completely multiplicative functions, also
knownasDirichletcharacters. Upperboundsforthemaximumsizeofthepartialsumsofthesefunctions
on intervals of positive integers is connected with the class number problem in algebraic number the-
ory, and with I.M. Vinogradov’s conjecture on the distribution of quadratic non-residues. By refining a
quantitativemeanvaluetheoremformultiplicativefunctions,wesignificantlyimprovetheexistingupper
bounds on the maximum size of partial sums of odd order Dirichlet characters, both unconditionally
and assuming the Generalized Riemann Hypothesis. We also show that our conditional results are best
possible unconditionally, up to a bounded power of loglogloglogq.
Regarding correlations, we prove a quantitative version of the bivariate Erd˝os-Kac theorem. That is,
ii
we show that the joint distribution of pairs of values of certain additive functions is asymptotically
an uncorrelated bivariate Gaussian, and find a quantitative error term in this approximation. We use
this probabilistic result to prove a theorem on the joint distribution of certain natural variants of the
M¨obius function at additively-related integers as a partial result in the direction of Chowla’s conjecture
on two-point correlations of the M¨obius function. We also apply our result to understanding the set of
pairs of consecutive integers with the same number of divisors.
A major theme in the thesis relates to how a multiplicative function can be rigidly characterized glob-
ally by certain local properties. As a first example, we show that a completely multiplicative function
that only takes finitely many values, vanishes at only finitely many primes and whose partial sums are
uniformlybounded,mustbeanon-principalDirichletcharacter. Thissolvesa60-year-oldopenproblem
of N.G. Chudakov. We also solve a folklore conjecture due to Elliott, Ruzsa and others on the gaps
between consecutive values of a unimodular completely multiplicative function, showing that these gaps
cannot be uniformly large. This is a corollary of several stronger results that are proved regarding the
distribution of consecutive values of multiplicative functions. For instance, we classify the set of all
unimodular completely multiplicative functions f such that {f(n)} is dense in T and for which the
n
sequence of pairs (f(n),f(n+1)) is dense in T2. In so doing, we resolve a conjecture of K´atai.
Finally, we make some progress on some natural variants of Chowla’s conjecture on sign patterns of the
Liouville function. In particular, we prove that certain natural collections of multiplicative functions
f :N→{−1,+1}aresuchthatthetuplesofvaluestheyproduceonalmost all 3-and4-termarithmetic
progressions equidistribute among all sign patterns of length 3 and 4, respectively.
Some of the aforementioned results are joint work with O. Klurman, or with Y. Lamzouri.
iii
Acknowledgements
I would like to first and foremost thank my thesis advisor John Friedlander for all of the time, energy
and moral and financial support that he generously provided me over the course of my time at U of T.
He has always been patient and understanding through my hardships, and enthusiastic about the work
that I have undertaken. His career advice has proved invaluable, and his constant encouragement has
helped me maintain my focus and drive throughout this journey.
I would like to thank the other members of my thesis committee, Professors Henry Kim and Jacob
Tsimerman, fortheirinputovertheyears, aswellasfortakingthetimetolookoverthecontentsofthis
thesis.
I would like to acknowledge the friendship and mentorship of Youness Lamzouri. Besides informing me
of the problem that would later become the subject of our joint work together, Youness provided me
withhelpfulguidanceinimprovingmywritingandpresentationskills,encouragedmeinvariousaspects
of my research and helped me learn how to think about problems. I gained a lot mathematically from
interacting with him on a frequent basis at the York University number theory seminar. His friendship
and generosity of time have been a major asset to me.
I would like to thank my friend and coauthor Oleksiy Klurman for making our process of collaboration
an enjoyable one. I owe him a debt of gratitude for introducing me to the subject of correlations, now a
major mathematical interest of mine; for the hours of mathematical discussions that we spent together,
whether in front of a blackboard or over the internet from different countries; for the many thought-
provoking problems that he shared with me; and for his unfailing sense of humour that never provided
for dull moments.
Severaloftheresultsinthisthesiscameaboutthroughhelpfulconversationswithvariouspeople,whom
I would like to acknowledge here. I am indebted to Sergei Konyagin and Terence Tao for introducing
Klurman and myself to Chudakov’s conjecture. I would also like to thank Imre K´atai for making us
aware of several of his conjectures, as well as his sustained interest in our work. In addition I want to
thankChristianElsholtzforhissuggestiontoworkonproblemsinvolvingsignpatternsofmultiplicative
functions. Finally, I would like to express gratitude to Maksym Radziwil(cid:32)(cid:32)l for his encouragement over
the years and for several helpful conversations.
I have benefited from the support of several institutions in fostering a productive work environment
outside of the University of Toronto. I would like to emphasize in particular my thanks to the Fields’
Institute, to the Mathematical Sciences Research Institute (MSRI), and to the Centre International
de Rencontres Math´emathiques (CIRM), where some important parts of the work of this thesis were
completed.
I would also like to acknowledge the tireless work of the administrative staff at the Department of
Mathematics. WithouttheinputofJemimaMerisca, SonjaInjac, PatrinaSeepersaudandeveryoneelse
who work behind the scenes to make our department run as seamlessly as it does, I would have had
many more (non-mathematical) headaches with which to contend.
I am grateful to the friends that helped make my time at U of T enjoyable. Most notably, I would like
to thank Asif Zaman for his constant encouragement and advice, his patience, his penchant for sanity
checks, andforshowingmewheretofindthebestrestaurantsnearcampus. Ialwaysfeltenlightenedby
our conversations. I would also like to thank Mateusz Olechnowicz for our stimulating and broadening
discussions. I hope that I was able to be as good a friend to them as they have been to me.
I would like to acknowledge my parents Libbie and Xavier for their constant, unending moral support,
iv
and for their unconditional love. I could not have accomplished any of what I have done in my life
without their input, in one way or another. I would also like to thank my brother Joshua and my
lifelong friend Michael Forbes for their irreplaceable friendship over the years.
I would like to thank my cats Milo and Stella for their affection and tenderness. I have found comfort
through them on countless occasions (though an unfortunate catnap on my keyboard did delay the
production of this thesis).
Finally, and most importantly, I would like to thank my wife and best friend Aran for everything. Her
supportandencouragementhavebeenimmeasurable. Shehasneverstoppedbelievinginme,evenwhen
Ihadseriousdoubtsaboutmyself, andhashelpedmedevelopincountlessways. Thisworkwouldnever
havematerializedintheformthatithaswithoutherpresenceinmylife. IhopeIhavemadeherproud.
v
Contents
0 Notation 1
1 Introduction 6
1.1 Background in Multiplicative and Pretentious Number Theory . . . . . . . . . . . . . . . 7
1.1.1 The Distribution of Primes and the M¨obius Function . . . . . . . . . . . . . . . . . 7
1.1.2 Mean Values of Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 Hal´asz’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.4 The Pretentious Distance and the Calculus of M . . . . . . . . . . . . . . . . . . . 12
1.2 Some Quantitative Extensions of Hal´asz’ Theorem . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Mean Values of Unbounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Logarithmically-Averaged Mean Value Theorems . . . . . . . . . . . . . . . . . . . 17
1.3 Partial Sums and Rigidity Theorems for Characters . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 The P´olya-Vinogradov Inequality for Odd Order Characters . . . . . . . . . . . . . 18
1.3.2 Rigid characterizations of Dirichlet characters . . . . . . . . . . . . . . . . . . . . . 20
1.4 Correlations of Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 The main conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 Truncated M¨obius functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.3 Some new results regarding sign patterns . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.4 Distributions of unimodular functions taking non-discrete values . . . . . . . . . . 29
1.4.5 Denseness and gap results for pairs of consecutive values . . . . . . . . . . . . . . . 30
1.5 Probabilistic Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.1 Strongly additive functions and the univariate Erd˝os-Kac theorem . . . . . . . . . 31
1.5.2 Asymptotic joint normality of additive functions . . . . . . . . . . . . . . . . . . . 32
1.5.3 Joint Poisson Laws for additive functions . . . . . . . . . . . . . . . . . . . . . . . 33
Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Joint Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 Quantitative Mean Value Theorems for Multiplicative Functions 37
2.1 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Upper and Lower Bounds for Non-Negative Multiplicative Functions . . . . . . . . . . . . 39
2.3 Decay Results for Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Asymptotic Formulae and Wirsing’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.1 Step i): Treatment of Large Primes. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4.2 Step ii): Treatment of Small Primes . . . . . . . . . . . . . . . . . . . . . . . . . . 64
vi
2.5 Asymptotic Formulae: The “Small Deviation” Case. . . . . . . . . . . . . . . . . . . . . . 70
2.6 Logarithmic Mean Values of Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . 76
3 Partial Sums and Rigidity Theorems for Dirichlet Characters 83
3.1 Dirichlet L-Functions and Character Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.1 Dirichlet Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.2 Character Sums and the P´olya-Vinogradov Inequality . . . . . . . . . . . . . . . . 85
3.1.3 Fixed Order Character Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2 Large Odd Order Character Sums and the P´olya-Vinogradov Inequality . . . . . . . . . . 88
3.2.1 A Lower Bound for M(χ): Proof of Theorem 3.2.8 . . . . . . . . . . . . . . . . . . 92
3.2.2 Estimates for the Distance D(χ,ψ;y): Proofs of Proposition 3.2.9 and Theorem
3.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.2.3 Estimates for D (y;T): Proofs of Propositions 3.2.4 and 3.2.5 . . . . . . . . . . . 101
χψ
3.2.4 Proofs of Theorems 3.2.6, 3.2.1 and 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . 107
3.3 On Chudakov’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.4 On a Variant of Cohn’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4 Correlations and Value Distribution of Multiplicative Functions 125
4.1 Correlations of Non-Pretentious Functions: Partial Results Toward Elliott’s Conjecture . 125
4.2 Correlations of Pretentious Functions: Klurman’s Method . . . . . . . . . . . . . . . . . . 127
4.3 Multilinear Averages of Pretentious Functions and Applications . . . . . . . . . . . . . . . 128
4.3.1 Application 1: Sign Changes of Multiplicative Functions in 3- and 4-term Arith-
metic Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.3.2 Application 2: Gowers Norms of 1-Bounded Multiplicative Functions . . . . . . . . 132
4.3.3 Detailed Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.3.4 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.3.5 Proof of Theorem 4.3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.3.6 Proof of Proposition 4.3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3.7 SignChangesofNon-PretentiousMultiplicativeFunctionson3-and4-termArith-
metic Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.3.8 Sign Patterns of Pretentious Functions in Almost All 4-term APs . . . . . . . . . . 157
4.4 Equidistribution and Denseness for Pairs of Consecutive Values . . . . . . . . . . . . . . . 168
4.5 On Consecutive Values of Unimodular Multiplicative Functions . . . . . . . . . . . . . . . 170
4.5.1 Preparatory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.5.2 Proof of Theorems 4.4.2 and 4.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.5.3 Proof of Theorem 4.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.6 Two-Dimensional Distribution of Unimodular Completely Multiplicative Functions at
Consecutive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.6.1 Auxiliary Results towards Theorem 4.6.4 . . . . . . . . . . . . . . . . . . . . . . . 184
4.6.2 Reduction to Pseudo-pretentious Functions . . . . . . . . . . . . . . . . . . . . . . 188
4.6.3 Localizing Values of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . 192
4.6.4 Towards the Case f(n)k =g(n)l =nit . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.6.5 Proof of Theorem 4.6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.6.6 The Eventually Rational Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
vii
4.6.7 Proof of Proposition 4.6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5 Joint Distributions of Additive Functions and Probabilistic Number Theory 200
5.1 Background in Classical Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.1.1 Strongly Additive Functions and the Univariate Erd˝os-Kac Theorem . . . . . . . . 202
5.1.2 Asymptotic Joint Normality of Additive Functions . . . . . . . . . . . . . . . . . . 204
5.2 On the Bivariate Erd˝os-Kac Theorem and Correlations of the M¨obius Function . . . . . . 205
5.2.1 A Quantitative Bivariate Erd˝os-Kac Theorem . . . . . . . . . . . . . . . . . . . . . 205
5.2.2 On the Binary Chowla Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
5.2.3 On Consecutive Values of the Divisor Functions. . . . . . . . . . . . . . . . . . . . 207
5.2.4 A Bivariate Kubilius Model and Multivariate Poisson Approximation. . . . . . . . 208
5.2.5 Uniform Approximation of φ by Normal and Poisson Characteristic Functions . . 216
x
5.2.6 Verifying that the Squarefree Integers form a Siftable Set . . . . . . . . . . . . . . 226
5.2.7 A Disjunction Theorem for Characteristic Functions . . . . . . . . . . . . . . . . . 230
5.2.8 Progress Towards the EPS Conjecture: Proof of Theorem 5.2.6 . . . . . . . . . . . 233
5.3 Quasi-Poisson Behaviour of Disjoint Subsets of the Primes and Restricted Prime Factors
of Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
5.3.1 Strategy of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
5.3.2 Auxiliary Lemmata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
5.3.3 Proof of Theorem 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
6 Appendix 246
6.1 Facts about Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.2 The Triangle Inequality for D and Some of its Basic Consequences . . . . . . . . . . . . . 249
6.3 Facts about Characters and Dirichlet L-functions . . . . . . . . . . . . . . . . . . . . . . . 252
6.4 Estimates for Prime Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.5 Some Basic Mean Value Theorems for Multiplicative Functions . . . . . . . . . . . . . . . 260
6.6 Chudakov’s Conjecture with α(cid:54)=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
6.7 Some Facts about Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 265
6.8 Some Results Regarding Probabilistic Number Theory . . . . . . . . . . . . . . . . . . . . 266
6.9 Miscellaneous Facts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Bibliography 271
viii
Chapter 0
Notation
In this section we outline the notational conventions that we shall use throughout the text.
General Conventions
• Givenf,g :C→Cwithg non-negative, wewritef(x)=O(g(x)), orequivalently, f(x)(cid:28)g(x), to
meanthatthereexistsanabsoluteconstantC >0andsomelargex ∈Rsuchthat|f(x)|≤Cg(x)
0
for all x≥x . In the special case that g(x)=1, this means that f is uniformly bounded for large
0
enough x.
Symmetrically, we write f(x)(cid:29)g(x), or equivalently f(x)=Ω(g(x)), if g(x)=O(f(x)).
• We write f(x)(cid:16)g(x) to mean that f(x)(cid:28)g(x)(cid:28)f(x).
• The implicit constants in expressions involving (cid:28),(cid:29),(cid:16) and O are absolute, unless implied other-
wisebythepresenceofoneormoresubscripts. Forinstance,givenaparameterα,f(x)=O (g(x)),
α
or equivalently f(x)(cid:28) g(x), if the constant implied by either of these relations is allowed to de-
α
pend on α.
• Given f and g as above, we write f(x) = o(g(x)) if f(x)/g(x) → 0 as x → ∞. When g(x) = 1,
this says that f(x) tends to 0 with x.
• We write f(x)∼g(x) if f(x)/g(x)−1=o(1).
• The letters a,b,d,k,l,m,n are reserved for integers, unless specified otherwise.
• The letters c,C will usually denote positive constants. Their meaning may change from one line
to the next without comment.
• The letters x,y,t,T,u,v are reserved for real numbers, unless specified otherwise.
• Thelettersz,w,swillusuallydenotecomplexnumbers. Thelettersr andρwilloftendenotetheir
moduli (though see the next subsection regarding ρ).
• The letter p will always refer to a prime, unless specified otherwise.
• The letters f and g will always be arithmetic functions, i.e., f,g :N→C. The letter h may serve
either as an arithmetic function, or as a positive integer, depending on the context.
1
Chapter 0. Notation 2
• Unlessotherwisespecified,χwillalwaysdenoteanon-principalDirichletcharacterwhosemodulus
is q. The symbol χ or χ(q) in case of ambiguity, will always denote a principal character modulo
0 0
q.
• For x∈R, we write e(x):=e2πix.
• Perhaps unconventionally, given z ∈ C we write arg(z) to be the element θ ∈ R/Z such that
z = |z|e(θ). We will identify arg(z) with an element in [0,1). This is, up to a factor of 2π,
equivalent to the principal branch of argument.
• The fractional part of a real number t will be denoted by {t}. The integer part of t will be written
as (cid:98)t(cid:99). We will also write (cid:107)t(cid:107):=min{|t−k|:k ∈Z}.
• Given x∈Ck, (cid:107)x(cid:107) will denote the (cid:96)k norm of x.
k
• U denotes the closed unit disc in the complex plane; T denotes the unit circle, i.e., T := {z ∈ U :
|z|=1}.
• Unless otherwise specified, the letters ε and η always refer to (possibly arbitrarily) small positive
real numbers. Their meaning may change from line to line without mention.
• Ifa,b∈Zthen(a,b)denotestheGCDofaandb(thoughthismaydependoncontext). Similarly,
[a,b] denotes the LCM of a and b.
• Given a,b∈N we write (a,b∞):=(cid:81) pνp(a).
p|b
• Given a property P of positive integers, we write 1 to be the function taking value 1 at integers
P
satisfying P, and 0 elsewhere.
• We will frequently use the following arithmetic functions:
(cid:80)
1. ω(n)= 1; this function is additive.
p|n
(cid:80)
2. Ω(n)= k; this function is completely additive.
pk||n
(cid:80)
3. φ(n)= 1 ; this function is multiplicative.
1≤a≤n (a,n)=1
(cid:80)
4. τ(n)= 1; this function is multiplicative.
d|n
5. More generally, if K ∈ N, we write τ (n) := (cid:80) 1; this function is multiplicative.
K d1···dK=n
Also, τ =τ with this notation.
2
6. λ(n)=(−1)Ω(n); this function is completely multiplicative.
7. µ(n) is the multiplicative function defined by µ(p)=λ(p) and µ(pk)=0 for all k ≥2 and all
primes p.
8. Λ(n) = logp if n = pk for some k and a prime p, otherwise Λ(n) = 0. This satisfies logn =
(cid:80)
Λ(d).
d|n
9. P+(n)isthemaximumprimefactordividingn; P−(n)istheminimumprimefactordividing
n.
10. P++(n):=max{pk :pk||n,k ≥1}; P−−(n):=min{pk :pk||n,k ≥1}.
• For y ≥2, S(y):={n∈N:P+(n)≤y}, and S∗(y):={n:P++(n)≤y}.