open problems about sumsets in finite abelian groups · 2016. 1. 25. · open problems about...
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Open problems about sumsetsin finite abelian groups
Bela Bajnok
January 5, 2016
Bela Bajnok Open problems about sumsets
Three types of sumsets
G : finite abelian group written additively, |G | = nA = {a1, . . . , am} ⊆ G , |A| = mh ∈ N
h-fold sumset of A:
hA = {Σmi=1λiai : (λ1, . . . , λm) ∈ Nm
0 , Σmi=1λi = h}
h-fold restricted sumset of A:
h A = {Σmi=1λiai : (λ1, . . . , λm) ∈ {0, 1}m, Σm
i=1λi = h}
h-fold signed sumset of A:
h±A = {Σmi=1λiai : (λ1, . . . , λm) ∈ Zm, Σm
i=1|λi | = h}
h A ⊆ hA ⊆ h±A Rarely equal
Bela Bajnok Open problems about sumsets
Three functions
ρ(G ,m, h) = min{|hA| : A ⊆ G , |A| = m}
ρ (G ,m, h) = min{|h A| : A ⊆ G , |A| = m}
ρ±(G ,m, h) = min{|h±A| : A ⊆ G , |A| = m}
ρ(G ,m, h) known; ρ (G ,m, h) and ρ±(G ,m, h) not fully known
ρ (G ,m, 1) = ρ(G ,m, 1) = ρ±(G ,m, 1) = m
ρ (G ,m, h) ≤ ρ(G ,m, h) ≤ ρ±(G ,m, h) Often equal
ρ (Z23, 4, 2) = 5, ρ(Z2
3, 4, 2) = 7, ρ±(Z23, 4, 2) = 8
Bela Bajnok Open problems about sumsets
ρ(G , m, h)
ρ(G ,m, h) = min{|hA| : A ⊆ G , |A| = m}
Cauchy, 1813; . . . ; Plagne, 2006⇓
ρ(G ,m, h) is known for all G , m, h
How to make hA small?G = Zn
Put A in a coset:
A ⊆ a + H ⇒ hA ⊆ h · a + H
Put A in an AP (arithmetic progression):
A ⊆ ∪k−1i=0 {a + i · g} ⇒ hA ⊆ ∪hk−h
i=0 {a + i · g}
Bela Bajnok Open problems about sumsets
ρ(G , m, h)
Put A in an AP of cosets:
Choose d ∈ D(n)
H ≤ Zn with |H| = d so H = ∪d−1j=0 {j · n/d}
Use dm/de cosets of H
m = cd + k with 1 ≤ k ≤ d
Ad = Ad(n,m) = ∪c−1i=0 (i + H)
⋃∪k−1
j=0 {c + j · n/d}
⇓hAd = ∪hc−1
i=0 (i + H)⋃∪hk−h
j=0 {hc + j · n/d}
⇓
|hAd | = min{n, hcd + min{d , hk − h + 1}}
Bela Bajnok Open problems about sumsets
ρ(G , m, h)
fd = fd(m, h) = hcd + d = (hdm/de − h + 1) · d
|hAd | = min{n, hcd + min{d , hk − h + 1}}= min{n, fd , hm − h + 1}= min{fn, fd , f1}
u(n,m, h) = min{fd(m, h) : d ∈ D(n)}
Theorem (Plagne)
For all G , m, and h ρ(G ,m, h) = u(n,m, h)
Proof:≤: construction as above but in any G≥: generalized Kneser’s Theorem
u(n,m, h) ∼ Hopf–Stiefel functionBela Bajnok Open problems about sumsets
ρ(G , m, h)—Inverse Problems
|A| = m, |hA| = ρ(G ,m, h) ⇒ A =?
Examples:
ρ(Z15, 6, 2) = 9⇓
A = (a1 + H) ∪ (a2 + H) with |H| = 3
ρ(Z15, 7, 2) = 13⇓
A = (a1 + H) ∪ (a2 + H) ∪ {a3} with |H| = 3 or
A = (a1 + H) ∪ {a2, a3} with |H| = 5 or
A = ∪6i=0{a + i · g}
Bela Bajnok Open problems about sumsets
ρ(G , m, h)—Inverse Problems
Special Case
p = min{p prime : p|n} and m ≤ p⇓
ρ(G ,m, h) = min{p, hm − h + 1}
Theorem (Kemperman)
hm − h + 1 < p and |hA| = ρ(G ,m, h) = hm − h + 1⇓h = 1 (and A arbitrary), or
A = AP
Conjecture
m ≤ p < hm − h + 1 and |hA| = ρ(G ,m, h) = p⇓
A ⊆ a + H with |H| = p
Bela Bajnok Open problems about sumsets
ρ (Zn, m, h)
ρ (G ,m, h) = min{|h A| : A ⊆ G , |A| = m}
How to make h A small? G = Zn
H ≤ G with |H| = d so H = ∪d−1j=0 {j · n/d}
m = cd + k with 1 ≤ k ≤ d
Ad = Ad(n,m) = ∪c−1i=0 (i + H)
⋃∪k−1
j=0 {c + j · n/d}
|hAd | = min{n, fd , hm − h + 1}
fd = fd(m, h) = hcd + d = (hdm/de − h + 1) · d
|h Ad | =
min{n, fd , hm − h2 + 1} if h ≤ min{k, d − 1}
min{n, hm − h2 + 1− δd} otherwise
δd = δd(m, h) is an explicitly computed “correction term”
Bela Bajnok Open problems about sumsets
ρ (Zn, m, h)
u (n,m, h) = min{|h Ad | : d ∈ D(n)}
ρ (Zn,m, h) ≤ u (n,m, h)
|h A1| = |h An| = min{n, hm − h2 + 1}
ρ (Zn,m, h) ≤ min{n, hm − h2 + 1}
Theorem (Dias da Silva, Hamidoune; Alon, Nathanson, Ruzsa)
p prime ⇒ ρ (Zp,m, h) = min{p, hm − h2 + 1}
For n ≤ 40,m, h:
ρ (Zn,m, h) =
u (n,m, h) more than 99.9% of time
u (n,m, h)− 1 otherwise
Bela Bajnok Open problems about sumsets
ρ (Zn, m, h)
m = k1 + (c − 1)d + k2 with 1 ≤ k1, k2 ≤ d , k1 + k2 > d
Bd = ∪k1−1j=0 {j ·n/d}
⋃∪c−1
i=1 (i ·g +H)⋃
∪k2−1j=0 {c ·g +(j0+ j)·n/d}
Bd is still in dm/de cosets of H but with ≤ 2 cosets not fully in Bd
|h Bd | < |h Ad | ⇔ n,m, h are very special
w (n,m, h) = min{|h Bd | : d ∈ D(n)}
ρ (Zn,m, h) ≤ min{u (n,m, h),w (n,m, h)}
Problem
ρ (Zn,m, h) = min{u (n,m, h),w (n,m, h)} ?
True for all n,m, h with n ≤ 40
Bela Bajnok Open problems about sumsets
ρ (Zn, m, 2)
For h = 2 this becomes:
Conjecture
ρ (Zn, m, 2) =
8>><>>:min{ρ(Zn, m, 2), 2m − 4} if 2|n and 2|m, or
(2m − 2)|n and log2(m − 1) 6∈ N;
min{ρ(Zn, m, 2), 2m − 3} otherwise.
Conjecture (Lev)
ρ (G ,m, 2) ≥ min{ρ(G ,m, 2), 2m − 3− |Ord(G , 2)|}
Theorem (Eliahou, Kervaire)
p odd prime ⇒ ρ (Zrp,m, 2) ≥ min{ρ(Zr
p,m, 2), 2m − 3}
Theorem (Plagne)
ρ (G ,m, 2) ≤ min{ρ(G ,m, 2), 2m − 2}
Bela Bajnok Open problems about sumsets
ρ (G , m, h)
Recall:
Special Case
p = min{p prime : p|n} and m ≤ p⇓
ρ(G ,m, h) = min{p, hm − h + 1}
Conjecture
p = min{p prime : p|n} and m ≤ p⇓
ρ (G ,m, h) = min{p, hm − h2 + 1}
Theorem (Karolyi)
Conjecture true for h = 2.
Bela Bajnok Open problems about sumsets
ρ (G , m, h)—Inverse Problems
Theorem (Kemperman)
hm − h + 1 < p and |hA| = ρ(G ,m, h) = hm − h + 1⇓h = 1 (and A arbitrary), or
A = AP
Conjecture
hm − h2 + 1 < p and |hA| = ρ (G ,m, h) = hm − h2 + 1⇓h ∈ {1,m − 1} (and A arbitrary),
h = 2, m = 4, and A = {a, a + g1, a + g2, a + g1 + g2}, or
A = AP
Theorem (Karolyi)
Conjecture true for h = 2.
Bela Bajnok Open problems about sumsets
ρ (G , m, h)—Inverse Problems
Conjecture
m ≤ p < hm − h + 1 and |hA| = ρ(G ,m, h) = p⇓
A ⊆ a + H with |H| = p
Conjecture
m ≤ p < hm − h2 + 1 and |hA| = ρ (G ,m, h) = p⇓
A ⊆ a + H with |H| = p
Bela Bajnok Open problems about sumsets
ρ±(G , m, h)
ρ±(G ,m, h) = min{|h±A| : A ⊆ G , |A| = m}
All work with Matzke, 2014-2015
Surprises:
ρ±(G ,m, h) depends on structure of G not just |G | = n
E.g. ρ±(Z23, 4, 2) = 8, ρ±(Z9, 4, 2) = 7
Usually |h±A| > |hA|, but often ρ±(G ,m, h) = ρ(G ,m, h)
E.g. n ≤ 24 ⇒ ρ±(G ,m, h) = ρ(G ,m, h) except ρ±(Z23, 4, 2)
Symmetric A (i.e. A = −A) is not always best
Sometimesnear-symmetric A (i.e. A \ {a} is symmetric) orasymmetric A (i.e. A ∩ −A = ∅)is best
But one of these three types will always yield ρ±(G ,m, h)
Bela Bajnok Open problems about sumsets
ρ±(G , m, h)
Theorem
For cyclic G ,ρ±(G ,m, h) = ρ(G ,m, h).
Proof: For each d ∈ D(n), find R ⊆ G so that
R is symmetric,
|R| ≥ m,
|h±R| = |hR| ≤ fd(m, h).
(A symmetric set A with |A| = m and |hA| ≤ fd(m, h) may notexist.)
Bela Bajnok Open problems about sumsets
ρ±(G , m, h)
Theorem
For G of type (n1, . . . , nr ),
ρ±(G ,m, h) ≤ u±(G ,m, h),
where
u±(G ,m, h) = min {Πρ±(Zni ,mi , h) : mi ≤ ni ,Πmi ≥ m}= min {Πu(ni ,mi , h) : mi ≤ ni ,Πmi ≥ m}= min{fd(m, h) : d ∈ D(G ,m)}
with
D(G ,m) = {d ∈ D(n) : d = Πdi , di ∈ D(ni ), dnr ≥ drm}
Note: for cyclic G , D(G ,m) = D(n).
Bela Bajnok Open problems about sumsets
ρ±(G , m, h)
Corollary
u(n,m, h) ≤ ρ±(G ,m, h) ≤ u±(G ,m, h),
where
u(n,m, h) = min{fd(m, h) : d ∈ D(n)}u±(G ,m, h) = min{fd(m, h) : d ∈ D(G ,m)}
Corollary
G is a 2-group ⇒ ρ±(G ,m, h) = ρ(G ,m, h)
Corollary
G is such that 6 ∃H ≤ G ,H ∼= Zrp, p > 2, r ≥ 2
⇓ρ±(G ,m, h) = ρ(G ,m, h)
Bela Bajnok Open problems about sumsets
ρ±(G , m, h)
Can we have ρ± (G ,m, h) < u±(G ,m, h)?
Proposition
d ∈ D(n) odd, d ≥ 2m + 1 ⇒ ρ± (G ,m, 2) ≤ d − 1.
Proof:
∃H ≤ G , |H| = d ,
∃A ⊆ H, |A| = m,A ∩ (−A) = ∅0 6∈ 2±A.
Conjecture
Do(n) = {d ∈ D(n) : d odd, d ≥ 2m + 1}.
ρ± (G ,m, h) = u±(G ,m, h) for all h ≥ 3.
ρ± (G ,m, 2) =
{u±(G ,m, 2) if Do(n) = ∅,min{u±(G ,m, 2), dm − 1} if dm = minD0(n)
Bela Bajnok Open problems about sumsets
ρ±(Zrp, m, h)
Elementary abelian groups Zrp with p odd prime
Theorem
p ≤ h ⇒ ρ±(Zrp,m, h) = ρ(Zr
p,m, h)
Theorem
h ≤ p − 1δ = 0 if h|p − 1 and δ = 1 if h 6 |p − 1k max s.t. pk + δ ≤ hm − h + 1c max s.t. (hc + 1) · pk + δ ≤ hm − h + 1
m ≤ (c + 1) · pk ⇒ ρ±(Zrp,m, h) = ρ(Zr
p,m, h)
Conjecture
m > (c + 1) · pk ⇒ ρ±(Zrp,m, h) > ρ(Zr
p,m, h)
Bela Bajnok Open problems about sumsets
ρ±(Z2p, m, 2)
Conjecture holds for r = 2 and h = 2:
Theorem
ρ±(Z2p,m, 2) = ρ(Z2
p,m, 2),m
m ≤ p,
m ≥ p2+12 , or
∃c ≤ p−12 s.t. c · p + p+1
2 ≤ m ≤ (c + 1) · p
Proof: Via results on critical pairs by Vosper, Kemperman, andLev.
Note:∣∣m : ρ±(Z2
p,m, 2) > ρ(Z2p,m, 2)
∣∣ = (p−1)2
4
Conjecture
|m : ρ±(G ,m, h) > ρ(G ,m, h)| < n4 for every G.
Bela Bajnok Open problems about sumsets
ρ±(Z2p, m, 2)
Theorem
m = cp + v with 0 ≤ c ≤ p − 1 and 1 ≤ v ≤ p⇓
c v ρ(Z2p, m, 2) ρ±(Z2
p, m, 2) u±(Z2p, m, 2)
v ≤ p−12
2m − 1 = 2m − 1 = 2m − 10
v ≥ p+12
p = p = p
v ≤ p−12
2m − 1 < (2c + 1)p = (2c + 1)p1 ≤ c ≤ p−3
2 v ≥ p+12
(2c + 1)p = (2c + 1)p = (2c + 1)p
v ≤ p−12
2m − 1 < p2 − 1 < p2
c = p−12 v ≥ p+1
2p2 = p2 = p2
c ≥ p+12
any v p2 = p2 = p2
The two boxed entries were proven by Lee.Bela Bajnok Open problems about sumsets
ρ±(G , m, h)—Inverse Problems
A(G ,m) = Sym(G ,m) ∪Nsym(G ,m) ∪Asym(G ,m) where
Sym(G ,m) = {A ⊆ G : |A| = m,A = −A}Nsym(G ,m) = {A ⊆ G : |A| = m,A \ {a} = −(A \ {a})}Asym(G ,m) = {A ⊆ G : |A| = m,A ∩ (−A) = ∅}
Theorem
ρ±(G ,m, h) = min{|h±A| : A ∈ A(G ,m)}
Note: each type is essential.
Problem
When is
ρ±(G ,m, h) = min{|h±A| : A ∈ Sym(G ,m)}?ρ±(G ,m, h) = min{|h±A| : A ∈ Nsym(G ,m)}?ρ±(G ,m, h) = min{|h±A| : A ∈ Asym(G ,m)}?
Bela Bajnok Open problems about sumsets
The h-critical number
χ(G , h) = min{m : A ⊆ G , |A| = m ⇒ hA = G}
χ±(G , h) = min{m : A ⊆ G , |A| = m ⇒ h±A = G}
χ (G , h) = min{m : A ⊆ G , |A| = m ⇒ h A = G}
χ(G , h) exists ∀G , h
χ±(G , h) exists ∀G , h
χ (G , h) existsm
h ∈ {1, n − 1}, ∀Gh ∈ {2, n − 2},G 6∼= Zr
2
3 ≤ h ≤ n − 3, ∀G
Bela Bajnok Open problems about sumsets
χ(G , h)
χ(G , h) = min{m : A ⊆ G , |A| = m ⇒ hA = G}
Theorem
χ(G , h) = v1(n, h) + 1
where vg (n, h) = max{(⌊
d−1−gcd(d ,g)h
⌋+ 1
)· n
d : d ∈ D(n)}
.
Theorem (Diamanda, Yap)
max{m : ∃A ⊆ Zn, |A| = m,A ∩ 2A = ∅} = v1(n, 3)
Theorem
max{m : ∃A ⊆ Zn, |A| = m,A ∩ 3A = ∅} = v2(n, 4)
Theorem (Hamidoune, Plagne)
k > l , gcd(k − l , n) = 1 ⇒max{m : ∃A ⊆ Zn, |A| = m, kA ∩ lA = ∅} = vk−l(n, k + l)
Bela Bajnok Open problems about sumsets
χ (G , h)
χ (G , h) = min{m : A ⊆ G , |A| = m ⇒ h A = G}
χ (G , 1) = χ (G , n − 1) = n
Proposition
G 6∼= Zr2 ⇒ χ (G , 2) = (n + |Ord(G , 2)|+ 3)/2
Proposition
(n + |Ord(G , 2)| − 1)/2 ≤ h ≤ n − 2 ⇒ χ (G , h) = h + 2
Problem
3 ≤ h ≤ (n + |Ord(G , 2)| − 3)/2 ⇒ χ (G , h) =?
Problem
3 ≤ h ≤ bn/2c − 1 ⇒ χ (Zn, h) =?
Bela Bajnok Open problems about sumsets
χ (G , h)
Theorem (Dias da Silva, Hamidoune; Alon, Nathanson, Ruzsa)
p prime ⇒ ρ (Zp,m, h) = min{p, hm − h2 + 1}
Corollary
p prime ⇒ χ (Zp, h) = b(p − 2)/hc+ h + 1
Theorem (Gallardo, Grekos, Habsieger, Hennecart, Landreau,Plagne)
n ≥ 12, even ⇒ χ (Zn, 3) = n/2 + 1
Theorem
n ≥ 12, even ⇒ χ (Zn, h) =
n/2 + 1 if h = 3, 4, . . . , n/2− 2;
n/2 + 2 if h = n/2− 1.
Bela Bajnok Open problems about sumsets
χ (Zn, 3)
Case 1: {d ∈ D(n) : d ≡ 2 (3)} 6= ∅, p smallestRecall:
Theorem
χ(G , 3) = v1(n, 3) + 1 =(1 + 1
p
)n3 + 1
Theorem
n ≥ 16 ⇒
χ (Zn, 3) ≥
(1 + 1
p
)n3 + 3 if n = p(
1 + 1p
)n3 + 2 if n = 3p(
1 + 1p
)n3 + 1 otherwise
Problem
χ (Zn, 3) = values above?
True for n prime, n even, n ≤ 50Bela Bajnok Open problems about sumsets
χ (Zn, 3)
Case 2: {d ∈ D(n) : d ≡ 2 (3)} = ∅Recall:
Theorem
χ(G , 3) = v1(n, 3) + 1 =⌊
n3
⌋+ 1
Theorem
n ≥ 11 ⇒
χ (Zn, 3) ≥
⌊
n3
⌋+ 4 if 9|n⌊
n3
⌋+ 3 otherwise
Problem
χ (Zn, 3) = values above?
True for n prime, n ≤ 50
Bela Bajnok Open problems about sumsets
The critical number
χ(G , N0) = min{m : A ⊆ G , |A| = m ⇒ ∪∞h=0hA = G}χ±(G , N0) = min{m : A ⊆ G , |A| = m ⇒ ∪∞h=0h±A = G}χ (G , N0) = min{m : A ⊆ G , |A| = m ⇒ ∪∞h=0h A = G}
p = min{d ∈ D(n) : d > 1}
∪∞h=0hA = ∪∞h=0h±A = 〈A〉⇓
χ(G , N0) = χ±(G , N0) = n/p + 1
Theorem (Dias Da Silva, Diderrich, Freeze, Gao, Geroldinger,Griggs, Hamidoune, Mann, Wou)
n ≥ 10 ⇒ χ (G , N0) =b2√
n − 2c+ 1 if G cyclic with n = p or n = pq whereq is prime, 3 ≤ p ≤ q ≤ p + b2
√p − 2c+ 1
n/p + p − 1 otherwise
Bela Bajnok Open problems about sumsets
χ (G , N0)—Inverse problems
A ⊆ Zn ↔ A ⊆ (−n/2, n/2]||A|| = Σa∈A|a|
Proposition
||A|| ≤ n − 2 ⇒ ∀b ∈ Zn,Σ(b · A) 6= Zn
Conjecture
p prime, A ⊆ Zp, |A| = χ (Zp, N0)− 1 = b2√
p − 2c⇓
ΣA 6= Zp ⇔ ∃b ∈ Zp \ {0}, ||b · A|| ≤ p − 2
Theorem (Nguyen, Szemeredi, Vu)
p prime, A ⊆ Zp, |A| ≥ 1.99√
p⇓
ΣA 6= Zp ⇒ ∃b ∈ Zp \ {0}, ||b · A|| ≤ p + O(√
p)
Bela Bajnok Open problems about sumsets
THANK YOU!
DANKE SCHON!
Bela Bajnok Open problems about sumsets