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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 Results Sketch of Proof Future Research
Introduction
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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
Results
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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
Overview
Behavior of the distribution of the number of missingsums.
The distribution has some strange behavior:
Not unimodal
Against missing certain number of sums.
18
Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
Question
Existence of DivotsFor a fixed different value of p, are there other divots?
Answer: Yes!
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Introduction Results Sketch of Proof Future Research
Question
Existence of DivotsFor a fixed different value of p, are there other divots?
Answer: Yes!
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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
Numerical analysis for different p ∈ (0,1) : p = 0.7
Figure: Distribution of |B| = k by simulating 1,000,000 subsets of{0,1,2, . . . ,400} with p = 0.7.
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Introduction Results Sketch of Proof Future Research
Numerical analysis for different p ∈ (0,1) : p = 0.8
Figure: Distribution of |B| = k by simulating 1,000,000 subsets of{0,1,2, . . . ,400} with p = 0.8.
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Introduction Results Sketch of Proof Future Research
Numerical analysis for different p ∈ (0,1) : p = 0.9
Figure: Distribution of |B| = k by simulating 1,000,000 subsets of{0,1,2, . . . ,400} with p = 0.9.
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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
Sketch of Proof
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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
Notation
Write S = L ∪M ∪ R, whereL ⊆ [0,29], M ⊆ [30,n − 31] and R ⊆ [n − 30,n − 1].
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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Introduction Results Sketch of Proof Future Research
Notation
Write S = L ∪M ∪ R, whereL ⊆ [0,29], M ⊆ [30,n − 31] and R ⊆ [n − 30,n − 1].
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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Introduction Results Sketch of Proof Future Research
Notation
Write S = L ∪M ∪ R, whereL ⊆ [0,29], M ⊆ [30,n − 31] and R ⊆ [n − 30,n − 1].
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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Introduction Results Sketch of Proof Future Research
Notation
Write S = L ∪M ∪ R, whereL ⊆ [0,29], M ⊆ [30,n − 31] and R ⊆ [n − 30,n − 1].
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
Our Bounds Are Sharp (p ≥ 0.7)
Figure:58
Introduction Results Sketch of Proof Future Research
Our Bounds Are Bad (p ≤ 0.6)
Figure:59
Introduction Results Sketch of Proof Future Research
Divot at 1
Figure:60
Introduction Results Sketch of Proof Future Research
Future Research
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Introduction Results Sketch of Proof Future Research
Question
Figure: Shift of Divots62
Introduction Results Sketch of Proof Future Research
Question
ConjectureThere are no divots at even numbers.
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Introduction Results Sketch of Proof Future Research
Question
Is there a value of p such that there are no divots?
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Introduction Results Sketch of Proof Future Research
Bibliography
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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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Introduction Results Sketch of Proof Future Research
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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