Open problems about sumsets
in finite abelian groups
Béla Bajnok
January 5, 2016
Béla Bajnok
Open problems about sumsets
Three types of sumsets
G : finite abelian group written additively, |G | = n
A = {a1 , . . . , am } ⊆ G , |A| = m
h∈N
h-fold sumset of A:
m
m
hA = {Σm
i=1 λi ai : (λ1 , . . . , λm ) ∈ N0 , Σi=1 λi = h}
h-fold restricted sumset of A:
m
m
hˆA = {Σm
i=1 λi ai : (λ1 , . . . , λm ) ∈ {0, 1} , Σi=1 λi = h}
h-fold signed sumset of A:
m
m
h± A = {Σm
i=1 λi ai : (λ1 , . . . , λm ) ∈ Z , Σi=1 |λi | = h}
hˆA ⊆ hA ⊆ h± A
Rarely equal
Béla Bajnok
Open problems about sumsets
Three functions
ρ(G , m, h) = min{|hA| : A ⊆ G , |A| = m}
ρˆ(G , m, h) = min{|hˆA| : A ⊆ G , |A| = m}
ρ± (G , m, h) = min{|h± A| : A ⊆ G , |A| = m}
ρ(G , m, h) known; ρˆ(G , m, h) and ρ± (G , m, h) not fully known
ρˆ(G , m, 1) = ρ(G , m, 1) = ρ± (G , m, 1) = m
ρˆ(G , m, h) ≤ ρ(G , m, h) ≤ ρ± (G , m, h)
Often equal
ρˆ(Z23 , 4, 2) = 5, ρ(Z23 , 4, 2) = 7, ρ± (Z23 , 4, 2) = 8
Béla 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):
hk−h
A ⊆ ∪k−1
i=0 {a + i · g } ⇒ hA ⊆ ∪i=0 {a + i · g }
Béla 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−1
j=0 {j · n/d}
Use dm/de cosets of H
m = cd + k with 1 ≤ k ≤ d
Ad = Ad (n, m) = ∪c−1
i=0 (i + H)
S
∪k−1
j=0 {c + j · n/d}
⇓
hAd =
∪hc−1
i=0 (i
+ H)
S
∪hk−h
j=0 {hc
+ j · n/d}
⇓
|hAd | = min{n, hcd + min{d, hk − h + 1}}
Béla 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 function
Béla 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 }
Béla 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
Béla Bajnok
Open problems about sumsets
ρˆ(Zn , m, h)
ρˆ(G , m, h) = min{|hˆA| : A ⊆ G , |A| = m}
How to make hˆA small? G = Zn
H ≤ G with |H| = d so H = ∪d−1
j=0 {j · n/d}
m = cd + k with 1 ≤ k ≤ d
[
c−1
Ad = Ad (n, m) = ∪i=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

if h ≤ min{k, d − 1}
 min{n, fd , hm − h2 + 1}
|hˆAd | =

min{n, hm − h2 + 1 − δd } otherwise
δd = δd (m, h) is an explicitly computed “correction term”
Béla Bajnok
Open problems about sumsets
ρˆ(Zn , m, h)
uˆ(n, m, h) = min{|hˆAd | : d ∈ D(n)}
ρˆ(Zn , m, h) ≤ uˆ(n, m, h)
|hˆA1 | = |hˆAn | = min{n, hm − h2 + 1}
ρˆ(Zn , m, h) ≤ min{n, hm − h2 + 1}
Theorem (Dias da Silva, Hamidoune; Alon, Nathanson, Ruzsa)
p prime ⇒ ρˆ(Zp , m, h) = min{p, hm − h2 + 1}
For n ≤ 40, m, h:
ρˆ(Zn , m, h) =


uˆ(n, m, h)

uˆ(n, m, h) − 1
Béla Bajnok
more than 99.9% of time
otherwise
Open problems about sumsets
ρˆ(Zn , m, h)
m = k1 + (c − 1)d + k2 with 1 ≤ k1 , k2 ≤ d, k1 + k2 > d
[
[ k −1
1 −1
2
Bd = ∪kj=0
{j ·n/d} ∪c−1
(i
·g
+H)
∪j=0
{c ·g +(j0 +j)·n/d}
i=1
Bd is still in dm/de cosets of H but with ≤ 2 cosets not fully in Bd
|hˆBd | < |hˆAd | ⇔ n, m, h are very special
wˆ(n, m, h) = min{|hˆBd | : d ∈ D(n)}
ρˆ(Zn , m, h) ≤ min{uˆ(n, m, h), wˆ(n, m, h)}
Problem
ρˆ(Zn , m, h) = min{uˆ(n, m, h), wˆ(n, m, h)} ?
True for all n, m, h with n ≤ 40
Béla Bajnok
Open problems about sumsets
ρˆ(Zn , m, 2)
For h = 2 this becomes:
Conjecture
ρˆ(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;
>
>
:
otherwise.
min{ρ(Zn , m, 2), 2m − 3}
Conjecture (Lev)
ρˆ(G , m, 2) ≥ min{ρ(G , m, 2), 2m − 3 − |Ord(G , 2)|}
Theorem (Eliahou, Kervaire)
p odd prime ⇒ ρˆ(Zrp , m, 2) ≥ min{ρ(Zrp , m, 2), 2m − 3}
Theorem (Plagne)
ρˆ(G , m, 2) ≤ min{ρ(G , m, 2), 2m − 2}
Béla Bajnok
Open problems about sumsets
ρˆ(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
⇓
ρˆ(G , m, h) = min{p, hm − h2 + 1}
Theorem (Károlyi)
Conjecture true for h = 2.
Béla Bajnok
Open problems about sumsets
ρˆ(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| = ρˆ(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 (Károlyi)
Conjecture true for h = 2.
Béla Bajnok
Open problems about sumsets
ρˆ(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| = ρˆ(G , m, h) = p
⇓
A ⊆ a + H with |H| = p
Béla 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
Sometimes
near-symmetric A (i.e. A \ {a} is symmetric) or
asymmetric A (i.e. A ∩ −A = ∅)
is best
But one of these three types will always yield ρ± (G , m, h)
Béla 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 not
exist.)
Béla 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 ≥ dr m}
Note: for cyclic G , D(G , m) = D(n).
Béla 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)
Béla 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.
u± (G , m, 2)
if Do (n) = ∅,
ρ± (G , m, 2) =
min{u± (G , m, 2), dm − 1} if dm = min D0 (n)
Béla 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(Zrp , m, h)
Theorem
h ≤p−1
δ = 0 if h|p − 1 and δ = 1 if h 6 |p − 1
k max s.t. p k + δ ≤ hm − h + 1
c max s.t. (hc + 1) · p k + δ ≤ hm − h + 1
m ≤ (c + 1) · p k ⇒ ρ± (Zrp , m, h) = ρ(Zrp , m, h)
Conjecture
m > (c + 1) · p k ⇒ ρ± (Zrp , m, h) > ρ(Zrp , m, h)
Béla Bajnok
Open problems about sumsets
ρ± (Z2p , m, 2)
Conjecture holds for r = 2 and h = 2:
Theorem
ρ± (Z2p , m, 2) = ρ(Z2p , m, 2),
m
m ≤ p,
p 2 +1
2 , or
p−1
≤ 2 s.t.
m≥
∃c
c ·p+
p+1
2
≤ m ≤ (c + 1) · p
Proof: Via results on critical pairs by Vosper, Kemperman, and
Lev.
2
Note: m : ρ± (Z2p , m, 2) > ρ(Z2p , m, 2) = (p−1)
4
Conjecture
|m : ρ± (G , m, h) > ρ(G , m, h)| <
Béla Bajnok
n
4
for every G .
Open problems about sumsets
ρ± (Z2p , m, 2)
Theorem
m = cp + v with 0 ≤ c ≤ p − 1 and 1 ≤ v ≤ p
⇓
ρ(Z2p , m, 2)
ρ± (Z2p , m, 2)
u± (Z2p , m, 2)
c
v
0
v≤
v≥
p−1
2
p+1
2
2m − 1
p
=
=
2m − 1
p
=
=
2m − 1
p
v≤
2m − 1
<
(2c + 1)p
=
(2c + 1)p
v≥
p−1
2
p+1
2
(2c + 1)p
=
(2c + 1)p
=
(2c + 1)p
v≤
v≥
p−1
2
p+1
2
2m − 1
p2
<
=
p2 − 1
p2
<
=
p2
p2
p2
=
p2
=
p2
1≤c≤
p−3
2
c=
p−1
2
c≥
p+1
2
any v
The two boxed entries were proven by Lee.
Béla 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)}?
Béla 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 }
χˆ(G , h) = min{m : A ⊆ G , |A| = m ⇒ hˆA = G }
χ(G , h) exists ∀G , h
χ± (G , h) exists ∀G , h
χˆ(G , h) exists
m
h ∈ {1, n − 1}, ∀G
h ∈ {2, n − 2}, G ∼
6 Zr
=
2
3 ≤ h ≤ n − 3, ∀G
Béla 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) + 1nj
k
d−1−gcd(d,g )
where vg (n, h) = max
+1 ·
h
n
d
o
: 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)
Béla Bajnok
Open problems about sumsets
χˆ(G , h)
χˆ(G , h) = min{m : A ⊆ G , |A| = m ⇒ hˆA = G }
χˆ(G , 1) = χˆ(G , n − 1) = n
Proposition
∼ Zr ⇒ χˆ(G , 2) = (n + |Ord(G , 2)| + 3)/2
G 6=
2
Proposition
(n + |Ord(G , 2)| − 1)/2 ≤ h ≤ n − 2 ⇒ χˆ(G , h) = h + 2
Problem
3 ≤ h ≤ (n + |Ord(G , 2)| − 3)/2 ⇒ χˆ(G , h) =?
Problem
3 ≤ h ≤ bn/2c − 1 ⇒ χˆ(Zn , h) =?
Béla Bajnok
Open problems about sumsets
χˆ(G , h)
Theorem (Dias da Silva, Hamidoune; Alon, Nathanson, Ruzsa)
p prime ⇒ ρˆ(Zp , m, h) = min{p, hm − h2 + 1}
Corollary
p prime ⇒ χˆ(Zp , h) = b(p − 2)/hc + h + 1
Theorem (Gallardo, Grekos, Habsieger, Hennecart, Landreau,
Plagne)
n ≥ 12, even ⇒ χˆ(Zn , 3) = n/2 + 1
Theorem
n ≥ 12, even ⇒ χˆ(Zn , h) =

 n/2 + 1

n/2 + 2
Béla Bajnok
if h = 3, 4, . . . , n/2 − 2;
if h = n/2 − 1.
Open problems about sumsets
χˆ(Zn , 3)
Case 1: {d ∈ D(n) : d ≡ 2 (3)} =
6 ∅, p smallest
Recall:
Theorem
χ(G , 3) = v1 (n, 3) + 1 = 1 + p1 n3 + 1
Theorem
n ≥ 16 ⇒  1 n

1
+

p 3 + 3 if n = p

 χˆ(Zn , 3) ≥
1 + p1 n3 + 2 if n = 3p



 1 + 1 n + 1 otherwise
p 3
Problem
χˆ(Zn , 3) = values above?
True for n prime, n even, n ≤ 50
Béla Bajnok
Open problems about sumsets
χˆ(Zn , 3)
Case 2: {d ∈ D(n) : d ≡ 2 (3)} = ∅
Recall:
Theorem
χ(G , 3) = v1 (n, 3) + 1 =
n
3
+1
Theorem
n ≥ 11 ⇒   n3 + 4 if 9|n
χˆ(Zn , 3) ≥
 n
3 + 3 otherwise
Problem
χˆ(Zn , 3) = values above?
True for n prime, n ≤ 50
Béla Bajnok
Open problems about sumsets
The critical number
χ(G , N0 ) = min{m : A ⊆ G , |A| = m ⇒ ∪∞
h=0 hA = G }
χ± (G , N0 ) = min{m : A ⊆ G , |A| = m ⇒ ∪∞
h=0 h± A = G }
χˆ(G , N0 ) = min{m : A ⊆ G , |A| = m ⇒ ∪∞
h=0 hˆA = G }
p = min{d ∈ D(n) : d > 1}
∞
∪∞
h=0 hA = ∪h=0 h± A = hAi
⇓
χ(G , N0 ) = χ± (G , N0 ) = n/p + 1
Theorem (Dias Da Silva, Diderrich, Freeze, Gao, Geroldinger,
Griggs, Hamidoune, Mann, Wou)
n ≥ 10

√⇒ χˆ(G , N0 ) =
 b2 n − 2c + 1 if G cyclic with n = p or n = pq√where
q is prime, 3 ≤ p ≤ q ≤ p + b2 p − 2c + 1

n/p + p − 1
otherwise
Béla Bajnok
Open problems about sumsets
χˆ(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| = χˆ(Zp , N0 ) − 1 = b2 p − 2c
⇓
ΣA 6= Zp ⇔ ∃b ∈ Zp \ {0}, ||b · A|| ≤ p − 2
Theorem (Nguyen, Szemerédi, Vu)
√
p prime, A ⊆ Zp , |A| ≥ 1.99 p
⇓
√
ΣA 6= Zp ⇒ ∃b ∈ Zp \ {0}, ||b · A|| ≤ p + O( p)
Béla Bajnok
Open problems about sumsets
THANK YOU!
DANKE SCHÖN!
Béla Bajnok
Open problems about sumsets