phase transitions in the distribution of missing sums and a powerful … · 2018. 10. 4. ·...
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Phase Transitions in the Distribution ofMissing Sums and a Powerful Family of
MSTD Sets
Steven J Miller (Williams College)Email: [email protected]
https://web.williams.edu/Mathematics/sjmiller/public_html/math/talks/talks.html
With Hung Viet Chu, Noah Luntzlara, Lily Shao, Victor Xu
INTEGERS Conference, Augusta, October 4, 2018
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Outline
Introduction to MSTD sets
Divot at 1
A powerful family of MSTD sets
Future research
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Introduction
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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!Conway (1969): {0,2,3,4,7,11,12,14}.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Martin and O’Bryant ’06
TheoremConsider In = {0,1, ...,n − 1}. The proportion of MSTDsubsets of In is bounded below by a positive constantc ≈ 2 · 10−7.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Results
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Distribution of the Number of Missing Sums (Uniform Model)
Figure: Frequency of the number of missing sums (q is probability ofnot choosing an element). The distribution is not unimodal. FromLazarev-Miller-O’Bryant.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Distribution of the Number of Missing Sums (Different Models)
Figure: Missing sums (q is probability of not choosing an element)from simulating 1,000,000 subsets of {0,1,2, ...,255}.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Sets of Missing Sums
Let In = {0,1,2, ...,n − 1}.
Form S ⊆ In randomly with probability p of picking anelement in In (q = 1− p: the probability of notchoosing an element).
Bn = (In + In)\(S + S) is the set of missing sums, |Bn|:the number of missing sums.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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, have divot at k ifP(|B| = k − 1) > P(|B| = k) < P(|B| = k + 1).
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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, have divot at k ifP(|B| = k − 1) > P(|B| = k) < P(|B| = k + 1).
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Example of Divot at 3
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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Lazarev-Miller-O’Bryant ’11
Divot at 7For p = 1/2, there is a divot at 7:P(|B| = 6) > P(|B| = 7) < P(|B| = 8).
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Question
Existence of DivotsFor a fixed different value of p, are there other divots?
Answer: Yes!
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Question
Existence of DivotsFor a fixed different value of p, are there other divots?
Answer: Yes!
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Numerical analysis for p = 0.7: divots at 1 and 3
Figure: Distribution of |B| = k by simulating 1,000,000 subsets of{0,1,2, . . . ,400} with p = 0.7.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Numerical analysis for different p = 0.8: divot at 1
Figure: Distribution of |B| = k by simulating 1,000,000 subsets of{0,1,2, . . . ,400} with p = 0.8.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Numerical analysis for different p = 0.9: divot at 1
Figure: Distribution of |B| = k by simulating 1,000,000 subsets of{0,1,2, . . . ,400} with p = 0.9.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Main Result
Divot at 1 [CLMSX’18]For p ≥ 0.68, there is a divot at 1:P(|B| = 0) > P(|B| = 1) < P(|B| = 2). Empirical evidencepredicts the value of p such that the divot at 1 starts toexist is between 0.6 and 0.7.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Sketch of Proof
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Fringe Analysis
Most of the missing sums come from the fringe: manymore ways to form middle elements than fringeelements.
Fringe analysis is enough to find good lower boundsand upper bounds for P(|B| = k).
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Fringe Analysis
Most of the missing sums come from the fringe: manymore ways to form middle elements than fringeelements.
Fringe analysis is enough to find good lower boundsand upper bounds for P(|B| = k).
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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 .
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Lower Bound
Given 0 ≤ k ≤ 30,
P(|B| = k) ≥k∑
i=0
[1− (a− 2)(qτ(L
ai ) + qτ(L
ak−i ))
− 1 + q(1− q)2 (q
min Lai + qmin La
k−i )
]P(La
i )P(Lak−i).
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Our Bounds Are Fairly Sharp (p ≥ 0.7)
Figure: We cannot see the blue line because our upper bound is sosharp that the orange line lies on the blue line.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Our Bounds Are Bad (p ≤ 0.6)
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Divot at 1
Figure: For p ≥ 0.68, the lower bounds for P(|B| = 0) and P(|B| = 2)are higher than the upper bound for P(|B| = 1). There is a divot at 1.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
A Powerful Family of MSTD sets
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Why Powerful?
Have appeared in the proof of many important resultsin previous works.
Give many sets with large log |A + A|/ log |A− A|.
Economically way to construct sets with fixed|A + A| − |A− A| (save more than four times of whatprevious construction has).
A is restricted-sum-dominant (RSD) if its restrictedsum set is bigger than its difference set. Improve thelower bound for the proportion of RSD sets from10−37 to 10−25.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
A different notation
We use a different notation to write a set; was firstintroduced by Spohn (1973).
Given a set S = {a1,a2, . . . ,an}, we arrange itselements in increasing order and find the differencesbetween two consecutive numbers to form asequence.
For example, S = {2,3,5,9,10}. We writeS = (2|1,2,4,1).
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
F family
F family
Let Mk denote 1,4, . . . ,4︸ ︷︷ ︸k -times
,3. Our family is
F : = {1,1,2,1,Mk1 ,Mk2 , . . . ,Mk` ,M1 : `, k1, . . . , k` ∈ N},
where M1 is either 1,1 or 1,1,2 or 1,1,2,1.
ConjectureAll sets in F are MSTD.
We proved that the conjecture holds for a periodic family.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Periodic Family [CLMS’18]
Sk ,` = (0|1,1,2,1,4, . . . ,4︸ ︷︷ ︸k -times
,3, . . . ,1,4, . . . ,4︸ ︷︷ ︸k -times
,3
︸ ︷︷ ︸`-times
,1,1,2,1)
has |Sk ,` + Sk ,`| − |Sk ,` − Sk ,`| = 2`.
S′k ,` = (0|1,1,2,1,4, . . . ,4︸ ︷︷ ︸k -times
,3, . . . ,1,4, . . . ,4︸ ︷︷ ︸k -times
,3
︸ ︷︷ ︸`-times
,1,1,2)
has |S′k ,` + S′k ,`| − |S′k ,` − S′k ,`| = 2`− 1.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
First Application
Sets A with fixed |A + A| − |A− A|Given x ∈ N, there exists a set A ⊆ [0,12 + 4x ] such that|A + A| − |A− A| = x . (Previous was [0,17x ]).
We save more than four times!
Method: Explicit constructions using Sk ,` and S′k ,`.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Second Application
Lower bound for restricted-sum-dominant setsFor n ≥ 81, the proportion of RSD subsets of{0,1,2, . . . ,n − 1} is at least 4.135 · 10−25. (Previous wasabout 10−37).
Method: Sk ,` reduces the needed fringe size from 120 to81.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Future Research
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Distribution of Missing Sums
Figure: Shift of Divots....45
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Future Research
Prove there are no divots at even numbers.
Is there a value of p such that there are no divots?
What about missing differences?
What if probability of choosing depends on n?
Work supported by NSF Grants DMS1561945 andDMS1659037, the Finnerty Fund, the University ofMichigan, Washington and Lee, and Williams College.
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets Future Research
Bibliography
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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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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Outline Introduction Results Sketch of Proof for Divot at 1 A Powerful Family of MSTD sets 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, 2653–2675.
S. J. Miller, B. Orosz and D. Scheinerman, Explicitconstructions of infinite families of MSTD sets, Journalof Number Theory 130 (2010), 1221–1233.
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