Given any \(n \in \mathbb{Z}_{\geq 0}\) and \(k \in [0,2n]\), we have \(|\{S : (S \subseteq [0,n]) \land (k \not \in S+S)\}| = 2^{|n-k|} \cdot 3^{\lfloor ((n+1)-|n-k|)/2 \rfloor}\). Nov 17 2021
Given any \(d \in \mathbb{Z}_{\geq 1}\) and \(n=2d^2-d\), we have \(\min \{ k : \exists A \exists B : (|A|=|B|=k) \land (A,B \subseteq [0,n]) \land (A+B = [0,2n]) \} = 2d\). Dec 13 2021
Given any \(d \in \mathbb{Z}_{\geq 1}\) and \(n=d^2-1\), we have \(\min \{ k : \exists A \exists B : (|A|=|B|=k) \land (A,B \subseteq [0,n]) \land (A+B \supseteq [0,n]) \} = d\). Dec 13 2021
Given any \(t \in \mathbb{Z}_{\geq 1}\), \(d=2^t\), and \(n=d^2-(t-1)d-2\), we have \(\min \{ k : \exists A \exists B : (|A|=|B|=k) \land (A,B \subseteq [0,n]) \land (A+_{\leq}B \supseteq [0,n]) \} = d\). Jan 08 2022
Given any \(m \in \mathbb{Z}_{\geq 2}\) and \(k_1, k_2 \in [0,m-1]\) with \(k_1 \not = k_2\) and \(\gcd(k_2 - k_1, m) = 1\), we have \(|\{S : (S \subseteq \mathbb{Z}_{m}) \land (k_1, k_2 \not \in S+S)\}| = F(m)\). Oct 29 2023
Given any \(m \in \mathbb{Z}_{\geq 0}\) and \(S \subseteq \mathbb{Z}_m\), if \(|S| \geq \lfloor m / 2 \rfloor + 1\), then \(S + S = S - S = \mathbb{Z}_m\). Nov 01 2023
Given any \(m \in \mathbb{Z}_{\geq 2}\) and \(k \in [1,m-1]\), we have \(|\{S : (S \subseteq \mathbb{Z}_{m}) \land (k \not \in S-S)\}| = L(m / \gcd(k,m))^{\gcd(k,m)}\). Nov 03 2023
Given any \(m \in \mathbb{Z}_{\geq 1}\) and \(k \in \mathbb{Z}_{m}\), we have \(|\{S : (S \subseteq \mathbb{Z}_{m}) \land (k \not \in S+S)\}| = 3^{\lfloor (m-1) / 2 \rfloor}\). Nov 03 2023
Given any \(d \in \mathbb{Z}_{\geq 2}\) and \(k=2d-2\), we have \(\max \{ n : \exists f : (f : \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}) \land ((i \in \mathbb{Z}_{\geq 0}) \rightarrow (f(i+2) - f(i+1) \geq f(i+1) - f(i) \geq 1)) \land (S = \{f(j) : j \in [0,k-1]\}) \land (S+S \supseteq [0,n]) \} = d^2 - 2\). Nov 26 2023
A108411 -- comment
Oct 05 2021
A137742 -- formula
Dec 09 2021
A309407 -- formula
Jan 17 2022
A000384 -- comment
Mar 09 2022
A000290 -- comment
Mar 09 2022
A014616 -- formula
Apr 28 2022
A000217 -- comment
May 04 2022
A167809 -- comment
May 16 2022
A066062 -- formula
Oct 15 2023
A191701 -- formula
Oct 27 2023
A196021 -- formula
Oct 27 2023
\([i,j]\) denotes the integer interval \(\{i,...,j\}\), or \(\{x \in \mathbb{Z} : i \leq x \leq j\}\), where \(i,j \in \mathbb{Z}\).
\(A+B\) denotes the sumset of \(A\) and \(B\), or \(\{a+b : (a \in A) \land (b \in B)\}\).
\(A+_{\leq}B\) denotes a partial sumset of \(A\) and \(B\), specifically, \(\{a+b : (a \in A) \land (b \in B)
\land (a \leq b)\}\).
\(A-B\) denotes the difference set of \(A\) and \(B\), or \(\{a-b : (a \in A) \land (b \in B)\}\).
\(F\), the sequence of Fibonacci numbers, is given by \(F(j) = F(j-1) + F(j-2)\), with \(F(0) = 0\) and \(F(1) =
1\).
\(L\), the sequence of Lucas numbers, is given by \(L(j) = L(j-1) + L(j-2)\), with \(L(2) = 3\) and \(L(3) = 4\).
finite additive 2-basis,
sumset,
additive basis,
additive combinatorics,
combinatorial number theory,
additive number theory,
integer interval,
additive doubling,
set,
subset,
superset,
cardinality,
element,
exponential function,
floor function,
nonnegative integer,
absolute value,
minimal,
optimal,
combinatorial covering problem
Thanks for reading!