Phase Transitions in the Distribution of Missing Sums › Mathematics › sjmiller › ...Young...

Introduction Results Sketch of Proof Future Research

Phase Transitions in the Distribution ofMissing Sums

Hung V. Chu (Washington and Lee University)Email: [email protected] Shao (Williams College)Email: [email protected]

In joint work with Noah Luntlaza, Steven J. Miller, Victor Xu.

Young Mathematicians ConferenceOhio State University

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Introduction

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Statement

A finite set of integers, |A| its size.

The sumset: A + A = {ai + aj |ai ,aj ∈ A}.

The difference set: A− A = {ai − aj |ai ,aj ∈ A}.

DefinitionA finite set of integers. A is called sum-dominated orMSTD (more-sum-than-difference) if |A + A| > |A− A|,balanced if |A + A| = |A− A| and difference-dominatedif |A + A| < |A− A|.

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Statement





4


Statement





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Statement





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False conjecture

Natural to think that |A + A| ≤ |A− A|.

Each pair (x , y), x 6= y gives two differences:x − y 6= y − x , but only one sum x + y .

However, sets A with |A + A| > |A− A| do exist!

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False conjecture




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False conjecture




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Examples

Conway: A1 = {0,2,3,4,7,11,12,14}

Marica: A2 = {0,1,2,4,7,8,12,14,15}

Pigarev and Freiman: A3 = {0,1,2,4,5,9,12,13,14,16,17,21,24,25,26,28,29}

Hegarty: A4 = {0,1,2,4,5,9,12,13,17,20,21,22,24,25,29,32,33,37,40,41,42,44,45}

Hegarty proved that the smallest cardinality of MSTDsets is 8.

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Examples

Conway: A1 = {0,2,3,4,7,11,12,14}

Marica: A2 = {0,1,2,4,7,8,12,14,15}

Pigarev and Freiman: A3 = {0,1,2,4,5,9,12,13,14,16,17,21,24,25,26,28,29}

Hegarty: A4 = {0,1,2,4,5,9,12,13,17,20,21,22,24,25,29,32,33,37,40,41,42,44,45}

Hegarty proved that the smallest cardinality of MSTDsets is 8.

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Martin and Obryant ’06

TheoremConsider In = {0,1, ...,n − 1}. The proportion of MSTDsubsets of In is bounded below by a positive constantc ≈ 2 · 10−7.

Later, Zhao improved the bound to 4.28 · 10−4 andproved that the limiting proportion exists.

Probabilistic method.

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Results

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Overview

Behavior of the distribution of the number of missingsums.

The distribution has some strange behavior:

Not unimodal

Against missing certain number of sums.

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Overview



Not unimodal


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Overview



Not unimodal


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Overview



Not unimodal


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Sets of Missing Sums

Let In = {0,1,2, ...,n − 1}.

Form S ⊆ In randomly with probability p of pickingan element in In. (q = 1− p: the probability of notchosing an element.)

Bn = (In + In)\(S + S) is the set of missing sums, |Bn|:the number of missing sums.

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Let In = {0,1,2, ...,n − 1}.



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Let In = {0,1,2, ...,n − 1}.



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Distribution of Missing Sums

Fix p ∈ (0,1), study P(|B| = k) = limn→∞ P(|Bn| = k).(Zhao proved that the limit exists.)

P(|B| = k): the limiting distribution of missing sums.

DivotFor some k ≥ 1, ifP(|B| = k − 1) > P(|B| = k) < P(|B| = k + 1), then thedistribution of sums has a divot at k .

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Example of Divot

Figure: Frequency of the number of missing sums for subsets of{0,1,2, ...,400} by simulating 1,000,000 subsets with p = 0.6.

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Why Divot?

Closely related to the behavior of missing sums in thesumset.

Many famous problems can be stated in the languageof sumsets: Goldbach’s Conjecture and Fermat’s LastTheorem.For example, let Pn be the set {1n,2n,3n, ...}. ThenFermat’s Last Theorem is equivalent to(Pn + Pn) ∩ Pn = ∅ for all n ≥ 3.

Interesting itself: two-bump distribution.

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Why Divot?

Closely related to the behavior of missing sums in thesumset.Many famous problems can be stated in the languageof sumsets: Goldbach’s Conjecture and Fermat’s LastTheorem.

For example, let Pn be the set {1n,2n,3n, ...}. ThenFermat’s Last Theorem is equivalent to(Pn + Pn) ∩ Pn = ∅ for all n ≥ 3.


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Why Divot?

Closely related to the behavior of missing sums in thesumset.Many famous problems can be stated in the languageof sumsets: Goldbach’s Conjecture and Fermat’s LastTheorem.For example, let Pn be the set {1n,2n,3n, ...}. ThenFermat’s Last Theorem is equivalent to(Pn + Pn) ∩ Pn = ∅ for all n ≥ 3.


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Why Divot?

Closely related to the behavior of missing sums in thesumset.Many famous problems can be stated in the languageof sumsets: Goldbach’s Conjecture and Fermat’s LastTheorem.For example, let Pn be the set {1n,2n,3n, ...}. ThenFermat’s Last Theorem is equivalent to(Pn + Pn) ∩ Pn = ∅ for all n ≥ 3.


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Numerical Analysis for p = 1/2

Figure: Frequency of the number of missing sums for all subsets of{0,1,2, ...,25}.

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Lazarev-Miller-O’Bryant ’11

Divot at 7For p = 1/2, there is a divot at 7, i.e.P(|B| = 6) > P(|B| = 7) < P(|B| = 8).

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Question

Existence of DivotsFor a fixed different value of p, are there other divots?

Answer: Yes!

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Question

Existence of DivotsFor a fixed different value of p, are there other divots?

Answer: Yes!

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Numerical analysis for different p ∈ (0,1) : p = 0.6

Figure: Distribution of |B| = k by simulating 1,000,000 subsets of{0,1,2, . . . ,400} with p = 0.6.

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Main Result

Chu-Luntzlara-Miller-Shao-XuFor p ≥ 0.68, there is a divot at 1, i.e.P(|B| = 0) > P(|B| = 1) < P(|B| = 2).

Empirical evidence predicts the value of p such thatthe divot at 1 starts to exist is between 0.6 and 0.7.

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Main Result

Chu-Luntzlara-Miller-Shao-XuFor p ≥ 0.68, there is a divot at 1, i.e.P(|B| = 0) > P(|B| = 1) < P(|B| = 2).

Empirical evidence predicts the value of p such thatthe divot at 1 starts to exist is between 0.6 and 0.7.

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Sketch of Proof

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Key Ideas

Want P(|B| = 0) > P(|B| = 1) < P(|B| = 2).

Establish an upper bound T 1 for P(|B| = 1).

Establish lower bounds T0 and T2 for P(|B| = 0) andP(|B| = 2), respectively.

Find values of p such that T2 > T 1 < T0.

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Key Ideas

Want P(|B| = 0) > P(|B| = 1) < P(|B| = 2).




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Key Ideas

Want P(|B| = 0) > P(|B| = 1) < P(|B| = 2).




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Key Ideas

Want P(|B| = 0) > P(|B| = 1) < P(|B| = 2).




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Fringe Analysis

Most of the missing sums come from the fringe: manymore ways to form middle elements than fringeelements.

Example: let S ⊆ {0,1, ...,10}. Then S + S ⊆ [0,20].Consider 0 = 0 + 0 and 20 = 10 + 10 while10 = 0 + 10 = 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5.

Fringe analysis is enough to find good lower boundsand upper bounds for P(|B| = k).

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Fringe Analysis




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Fringe Analysis




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Setup

Consider S ⊆ {0,1,2, ...,n − 1} with probability p ofeach element being picked.

Analyze fringe of size 30.

Write S = L ∪M ∪ R, whereL ⊆ [0,29], M ⊆ [30,n − 31] and R ⊆ [n − 30,n − 1].

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Setup




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Setup




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Notation


Lk : the event that L + L misses k sums in [0,29].

Lak : the event that L + L misses k sums in [0,29] and

contains [30,48].

Similar notations applied for R.

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Notation




contains [30,48].


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Notation




contains [30,48].


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Notation




contains [30,48].


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Upper Bound

Given 0 ≤ k ≤ 30,

P(|B| = k) ≤k∑

i=0

P(Li)P(Lk−i) +2(2q − q2)15(3q − q2)

(1− q)2 .

(1)

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Lower Bound

Given 0 ≤ k ≤ 30,

P(|B| = k) ≥k∑

i=0

[1− (a− 2)(qτ(La

i ) + qτ(Lak−i ))

− 1 + q(1− q)2 (q

min Lai + qmin La

k−i )

]P(La

i )P(Lak−i).

(2)

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Our Bounds Are Sharp (p ≥ 0.7)

Figure:58


Our Bounds Are Bad (p ≤ 0.6)

Figure:59


Divot at 1

Figure:60


Future Research

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Question

Figure: Shift of Divots62


Question

ConjectureThere are no divots at even numbers.

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Question

Is there a value of p such that there are no divots?

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Bibliography

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Bibliography

O. Lazarev, S. J. Miller, K. O’Bryant, Distribution ofMissing Sums in Sumsets (2013), ExperimentalMathematics 22, no. 2, 132–156.

P. V. Hegarty, Some explicit constructions of sets withmore sums than differences (2007), Acta Arithmetica130 (2007), no. 1, 61–77.

P. V. Hegarty and S. J. Miller, When almost all sets aredifference dominated, Random Structures andAlgorithms 35 (2009), no. 1, 118–136.

G. Iyer, O. Lazarev, S. J. Miller and L. Zhang,Generalized more sums than differences sets, Journalof Number Theory 132 (2012), no. 5, 1054–1073.

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Bibliography

J. Marica, On a conjecture of Conway, Canad. Math.Bull. 12 (1969), 233–234.

G. Martin and K. O’Bryant, Many sets have moresums than differences, in Additive Combinatorics,CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc.,Providence, RI, 2007, pp. 287–305.

M. Asada, S. Manski, S. J. Miller, and H. Suh, Fringepairs in generalized MSTD sets, International Journalof Number Theory 13 (2017), no. 10, 2653a2675.

S. J. Miller, B. Orosz and D. Scheinerman, Explicitconstructions of infinite families of MSTD sets, Journal ofNumber Theory 130 (2010), 1221–1233.

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Bibliography

S. J. Miller and D. Scheinerman, Explicit constructionsof infinite families of mstd sets, Additive NumberTheory, Springer, 2010, pp. 229-248.

S. J. Miller, S. Pegado and L. Robinson, ExplicitConstructions of Large Families of Generalized MoreSums Than Differences Sets, Integers 12 (2012),#A30.

M. B. Nathanson, Problems in additive number theory,1, Additive combinatorics, 263–270, CRM Proc.Lecture Notes 43, Amer. Math. Soc., Providence, RI,2007.

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Bibliography

M. B. Nathanson, Sets with more sums thandifferences, Integers : Electronic Journal ofCombinatorial Number Theory 7 (2007), Paper A5(24pp).

Y. Zhao, Constructing MSTD sets using bidirectionalballot sequences, Journal of Number Theory 130(2010), no. 5, 1212–1220.

Y. Zhao, Sets characterized by missing sums anddifferences, Journal of Number Theory 131 (2011),no. 11, 2107–2134.

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THANK YOU!

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