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$G$ is a Group. Ex: 4 is $\mathbb{Z}_4$ and 2x4 is $\mathbb{Z}_2\times\mathbb{Z}_4$.

Signed (${\ }_{\pm}$)
Restricted ($\ \hat{\ }\ $)

Functions and Notation

Chapter A: Maximum sumset size

$$\nu_\Lambda(G,m,H)= \max \{ |H_\Lambda A| \mid A \subseteq G, |A|=m\}$$

The $\nu$ function is defined so that $\nu_\Lambda(G, m, H)$ is the largest size of $H_\Lambda A,$ where $|A| = m.$ In other words, $\nu(G, m, h)$ is the largest the $H$-fold sumset of a size $m$ subset of $G$ can be.

Note that we have a relation (see Proposition A.9)

$\nu(G, m, [0, s]) = \nu(G, m + 1, s).$

A.1: Unrestricted sumsets

$\nu(G,m,h)= \max \{ |hA| \mid A \subseteq G, |A|=m\}$
$\nu(G,m,[0,s])= \max \{ |[0,s]A| \mid A \subseteq G, |A|=m\}$
$\nu(G,m,\mathbb{N}_0)= \max \{ |\langle A \rangle | \mid A \subseteq G, |A|=m\}$

A.2: Unrestricted signed sumsets

$\nu_{\pm} (G,m,h)= \max \{ |h_{\pm} A| \mid A \subseteq G, |A|=m\}$
$\nu_{\pm} (G,m,[0,s])= \max \{ |[0,s]_{\pm} A| \mid A \subseteq G, |A|=m\}$
$\nu_{\pm} (G,m,\mathbb{N}_0)= \max \{ |\langle A \rangle | \mid A \subseteq G, |A|=m\}$

A.3: Restricted sumsets

$\nu \hat{\;} (G,m,h)= \max \{ |h\hat{\;} A| \mid A \subseteq G, |A|=m\}$
$\nu\hat{\;} (G,m,[0,s])= \max \{|[0,s]\hat{\;} A| \mid A \subseteq G, |A|=m\}$
$\nu \hat{\;} (G,m,\mathbb{N}_0)= \max \{ |\Sigma A | \mid A \subseteq G,|A|=m\}$

A.4: Restricted signed sumsets

$\nu \hat{_{\pm}} (G,m,h)= \max \{ |h \hat{_{\pm}} A| \mid A \subseteq G, |A|=m\}$
$\nu \hat{_{\pm}} (G,m,[0,s])= \max \{ |[0,s] \hat{_{\pm}} A| \mid A \subseteq G, |A|=m\}$
$\nu \hat{_{\pm}} (G,m,\mathbb{N}_0)=\max \{ |\Sigma _{\pm} A | \mid A \subseteq G, |A|=m\}$

Chapter B: Spanning sets

$$\phi_\Lambda(G,H)= \min \{ |A| \mid A \subseteq G, H_\Lambda A=G\}$$

The phi function is defined so that $\phi_\Lambda(G,H)$ is the minimum size of a spanning set of $G$. A spanning set is a set $A$ so that the $H$-fold sumset of $A$ is the entire group $G$.

B.1: Unrestricted sumsets

$\phi(G,h)= \min \{ |A| \mid A\subseteq G, hA=G\}$
$\phi(G,[0,s])= \min \{ |A| \mid A \subseteq G, [0,s]A=G\}$
$\phi(G,\mathbb{N}_0)= \min \{ |A | \mid A\subseteq G, \langle A \rangle=G\}$

B.2: Unrestricted signed sumsets

$\phi_{\pm} (G,h)= \min \{ | A| \mid A \subseteq G,h_{\pm} A=G\}$
$\phi_{\pm} (G,[0,s])= \min \{ | A| \mid A \subseteq G, [0,s]_{\pm} A=G\}$
$\phi_{\pm} (G,\mathbb{N}_0)=\min \{ |A | \mid A \subseteq G, \langle A \rangle=G\}$

B.3: Restricted sumsets

$\phi \hat{\;} (G,h)= \min \{ | A| \mid A\subseteq G, h\hat{\;} A=G\}$
$\phi\hat{\;} (G,[0,s])= \min \{ |A| \mid A \subseteq G, [0,s]\hat{\;} A = G\}$
$\phi \hat{\;}(G,\mathbb{N}_0)= \min \{ | A | \mid A \subseteq G, \Sigma A =G\}$

B.4: Restricted signed sumsets

$\phi \hat{_{\pm}} (G,h)=\min \{ | A| \mid A \subseteq G, h \hat{_{\pm}} A = G\}$
$\phi\hat{_{\pm}} (G,[0,s])= \min \{ | A| \mid A \subseteq G, [0,s]\hat{_{\pm}} A = G\}$
$\phi \hat{_{\pm}} (G,\mathbb{N}_0)= \min\{ | A | \mid A \subseteq G, \Sigma _{\pm} A = G\}$

Chapter C: Sidon sets

$$\sigma_\Lambda(G,H)=\max\{|A|\mid A\subseteq G, |H_\Lambda A|=|\Lambda^{|A|}(H)|\}$$

The $\sigma$ function is defined so that $\sigma_\Lambda(G,H)$ is the maximum size of a sidon set of $G$. A sidon set is defined precisely in Chapter C of Bela's book.

C.1: Unrestricted sumsets

$\sigma(G,h)= \max \left \{ |A| \mid A\subseteq G, |hA|=|\mathbb{N}_0^m (h)|={m+h-1 \choose h}\right \}$

$\sigma(G,[0,s])= \max \left \{ |A| \mid A \subseteq G,|[0,s]A|=\Sigma_{h=0}^s |\mathbb{N}_0^m (h)|={m+s \choose s} \right\}$

C.2: Unrestricted signed sumsets

$\sigma_{\pm} (G,h)= \max\left \{ | A| \mid A \subseteq G, |h_{\pm} A|=|\mathbb{Z}^m(h)|=c(h,m)\right \}$

$\sigma_{\pm} (G,[0,s])= \max \left \{ | A|\mid A \subseteq G, |[0,s]_{\pm} A|=\Sigma_{h=0}^s |\mathbb{Z}^m(h)|=a(m,s) \right \}$

C.3: Restricted sumsets

$\sigma \hat{\;}(G,h)= \max \left \{ | A| \mid A \subseteq G, |h\hat{\;}A|=|\hat{\mathbb{N}}_0^m (h)|={m \choose h}\right \}$

$\sigma\hat{\;} (G,[0,s])= \max \left \{ | A| \mid A \subseteq G,|[0,s]\hat{\;} A| = \Sigma_{h=0}^s |\hat{\mathbb{N}}_0^m (h)|=\Sigma_{h=0}^s {m \choose h} \right \}$

$\sigma \hat{\;}(G,\mathbb{N}_0)= \max \left \{ | A | \mid A \subseteq G, |\Sigma A|= \Sigma_{h=0}^m |\hat{\mathbb{N}}_0^m (h)|= 2^m \right \}$

C.4:Restricted signed sumsets

$\sigma \hat{_{\pm}} (G,h)= \max \left\{ | A| \mid A \subseteq G, |h \hat{_{\pm}} A| =|\hat{\mathbb{Z}}^m (h)|={m \choose h} 2^h\right \}$

$\sigma\hat{_{\pm}} (G,[0,s])= \max \left \{ | A| \mid A \subseteq G,|[0,s] \hat{_{\pm}} A| = \Sigma_{h=0}^s |\hat{\mathbb{Z}}^m (h)|=\Sigma_{h=0}^s {m \choose h} 2^h \right \}$

$\sigma \hat{_{\pm}}(G,\mathbb{N}_0)= \max \left \{ | A | \mid A \subseteq G, |\Sigma_{\pm} A |=\Sigma_{h=0}^m |\hat{\mathbb{Z}}^m (h)|= 3^m \right \}$

Chapter D: Minimum sumset size

$$\rho_\Lambda(G,m,G)=\min\{|H_\Lambda A| \mid A\subseteq G, |A|=m\}$$

The $\rho$ function is defined so that $\rho_\Lambda(G, m, H)$ is the smallest size of $H_\Lambda A$, where $|A| = m$. In other words, $\rho_\Lambda(G, m, H)$ is the smallest the $H$-fold $\Lambda$ sumset of a size $m$ subset of $G$ can be.

D.1: Unrestricted sumsets

$\rho(G,m,h)= \min \{ |hA| \mid A\subseteq G, |A|=m\}$
$\rho(G,m,[0,s])= \min \{ |[0,s]A| \mid A\subseteq G, |A|=m\}$
$\rho(G,m,\mathbb{N}_0)= \min \{ |\langle A\rangle | \mid A \subseteq G, |A|=m\}$

D.2: Unrestricted signed sumsets

$\rho_{\pm} (G,m,h)= \min \{ |h_{\pm} A| \mid A \subseteq G, |A|=m\}$
$\rho_{\pm} (G,m,[0,s])= \min \{ |[0,s]_{\pm} A| \mid A \subseteq G, |A|=m\}$
$\rho_{\pm} (G,m,\mathbb{N}_0)= \min \{|\langle A \rangle | \mid A \subseteq G, |A|=m\}$

D.3: Restricted sumsets

$\rho \hat{\;} (G,m,h)= \min \{ |h\hat{\;} A| \mid A\subseteq G, |A|=m\}$
$\rho\hat{\;} (G,m,[0,s])= \min \{|[0,s]\hat{\;} A| \mid A \subseteq G, |A|=m\}$
$\rho \hat{\;}(G,m,\mathbb{N}_0)= \min \{ |\Sigma A | \mid A \subseteq G,|A|=m\}$

D.4: Restricted signed sumsets

$\rho \hat{_{\pm}}(G,m,h)= \min \{ |h \hat{_{\pm}} A| \mid A \subseteq G, |A|=m\}$
$\rho \hat{_{\pm}} (G,m,[0,s])= \min \{ |[0,s] \hat{_{\pm}} A|\mid A \subseteq G, |A|=m\}$
$\rho \hat{_{\pm}}(G,m,\mathbb{N}_0)= \min \{ |\Sigma _{\pm} A | \mid A \subseteq G,|A|=m\}$

Chapter E: The critical number

$$\chi_\Lambda(G,h)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow H_\Lambda A=G \}$$

The $\chi$ function is defined so that $\chi_\Lambda(G, H)$ is the smallest $m$ for which every $m$ size subset of $G$ spans $G$.

E.1: Unrestricted sumsets

$\chi(G,h)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow hA=G \}$
$\chi(G,[0,s])= \min \{ m\mid A \subseteq G, |A|=m \Rightarrow [0,s]A=G \}$
$\chi(G,\mathbb{N}_0)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow \langle A \rangle =G \}$

E.2: Unrestricted signed sumsets

$\chi_{\pm} (G,h)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow h_{\pm}A=G \}$
$\chi_{\pm} (G,[0,s])= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow [0,s]_{\pm}A=G \}$
$\chi_{\pm} (G,\mathbb{N}_0)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow \langle A \rangle=G \}$

E.3: Restricted sumsets

$\chi \hat{\;} (G,h)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow h\hat{\;}A=G \}$
$\chi\hat{\;} (G,[0,s])= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow [0,s] \hat{\;}A=G \}$
$\chi \hat{\;} (G,\mathbb{N}_0)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow \Sigma A=G \}$

E.4: Restricted signed sumsets

$\chi \hat{_{\pm}} (G,h)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow h\hat{_{\pm}} A=G \}$
$\chi \hat{_{\pm}} (G,[0,s])= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow [0,s] \hat{_{\pm}} A=G \}$
$\chi \hat{_{\pm}} (G,\mathbb{N}_0)= \min \{ m \mid A \subseteq G, |A|=m \Rightarrow \Sigma \hat{_{\pm}} A=G \}$

Chapter F: Zero-sum-free sets

$$\tau_\Lambda(G, H) = \max\{|A| \mid A\subseteq G, 0\not\in H_\Lambda A\}.$$

The $\tau$ function is defined so that $\tau_\Lambda(G, H)$ is the maximum size of a zero-$H$-free sumset. In other words, it's the largest size of a set $A$ such that $HA$ does not contain 0.

F.1: Unrestricted sumsets

$\tau(G,h)= \max \{ |A| \mid A \subseteq G, 0 \not \in hA\}$
$\tau(G,[1,t])= \max \{ |A| \mid A \subseteq G, 0 \not \in [1,t]A\}$

F.2: Unrestricted signed sumsets

$\tau_{\pm} (G,h)= \max \{ | A| \mid A \subseteq G, 0 \not \in h_{\pm} A\}$
$\tau_{\pm} (G,[1,t])= \max \{ | A| \mid A \subseteq G, 0 \not \in [1,t]_{\pm} A\}$

F.3: Restricted sumsets

$\tau \hat{\;} (G,h)= \max \{ | A| \mid A \subseteq G, 0 \not \in h\hat{\;} A\}$
$\tau\hat{\;} (G,[1,t])= \max \{ | A| \mid A \subseteq G, 0 \not \in [1,t]\hat{\;} A \}$
$\tau \hat{\;} (G,\mathbb{N})= \max \{ | A | \mid A \subseteq G, 0 \not \in [1,m]\hat{\;} A \}$

F.4: Restricted signed sumsets

$\tau \hat{_{\pm}} (G,h)= \max \{ | A| \mid A \subseteq G, 0 \not \in h \hat{_{\pm}} A \}$
$\tau \hat{_{\pm}} (G,[1,t])= \max \{ | A| \mid A \subseteq G, 0 \not \in [1,t] \hat{_{\pm}} A \}$
$\tau \hat{_{\pm}} (G,\mathbb{N})= \max \{ | A | \mid A \subseteq G, 0 \not \in [1,m] \hat{_{\pm}} A \}$

Chapter G: Sum-free sets

$$\mu_\Lambda(G,\{k,l\})=\max\{|A|\mid A\subseteq G, k_\Lambda A\cap l_\Lambda A = \emptyset\}$$

The $\mu$ function is defined so that $\mu_\Lambda(G, \{k, l\})$ is the maximum size of a set $A$ such that $k_\Lambda A$ is disjoint from $l_\Lambda A$.

G.1: Unrestricted sumsets

$\mu(G,\{k,l\})= \max \{ |A| \mid A \subseteq G, kA \cap lA = \emptyset\}$
$\mu(G,[0,s])= \max \{ |A| \mid A \subseteq G, 0 \leq l < k \leq s \Rightarrow kA \cap lA = \emptyset\}$

G.2: Unrestricted signed sumsets

$\mu_{\pm} (G,\{k,l\})= \max \{ | A| \mid A \subseteq G, k_{\pm}A \cap l_{\pm}A = \emptyset\}$
$\mu_{\pm} (G,[0,s])= \max \{ | A| \mid A \subseteq G, 0 \leq l < k \leq s \Rightarrow k_{\pm}A \cap l_{\pm}A = \emptyset\}$

G.3: Restricted sumsets

$\mu \hat{\;} (G,\{k,l\})= \max \{ | A| \mid A \subseteq G, k\hat{\;}A \cap l\hat{\;}A = \emptyset\}$
$\mu\hat{\;} (G,[0,s])= \max \{ | A| \mid A \subseteq G, 0 \leq l < k \leq s \Rightarrow k\hat{\;}A \cap l\hat{\;}A = \emptyset \}$
$\mu \hat{\;} (G,\mathbb{N}_0)= \max \{ | A | \mid A \subseteq G, 0 \leq l < k \Rightarrow k \hat{\;} A \cap l \hat{\;} A = \emptyset \}$

G.4: Restricted signed sumsets

$\mu \hat{_{\pm}} (G,\{k,l\})= \max \{ | A| \mid A \subseteq G, kA \cap lA = \emptyset \}$
$\mu \hat{_{\pm}} (G,[0,s])= \max \{ | A| \mid A \subseteq G, 0 \leq l < k \leq s \Rightarrow k\hat{_{\pm}}A \cap l\hat{_{\pm}}A = \emptyset \}$
$\mu \hat{_{\pm}} (G,\mathbb{N}_0)= \max \{ | A | \mid A \subseteq G, 0 \leq l < k \Rightarrow k\hat{_{\pm}}A \cap l\hat{_{\pm}}A = \emptyset \}$