Suppose $A = (a_n) = (a_1, a_2, a_3, . . .)$ is an positive, increasing sequence of integers.

Define an $A$- expressible number $c$ if $c$ is the alternating sum of a finite subsequence of $A.$ To form such a sum, choose a finite subset of the sequence $A,$ list those numbers in increasing order (no repetitions allowed), and combine them with alternating plus and minus signs. We allow the trivial case of one-element subsequences, so that each an is $A-$expressible.

Definition. Sequence $A = (a_n)$ is an “alt-basis” if every positive integer is uniquely $A-$ expressible. That is, for every integer $m > 0,$ there is exactly one way to express $m$ as an alternating sum of a finite subsequence of $A.$

Examples. Sequence $B = (2^{n−1}) = (1, 2, 4, 8, 16, . . .)$ is not an alt-basis because some numbers are B-expressible in more than one way. For instance $3 = −1 + 4 = 1 − 2 + 4.$

Sequence $C = (3^{n−1}) = (1, 3, 9, 27, 81, . . .)$ is not an alt-basis because some numbers (like 4 and 5) are not C-expressible.

An example of an alt-basis is $\{2^n-1\}=\{1,3,7,15,31,\ldots\}$

Is there a fairly simple test to determine whether a given sequence is an alt basis?

I have attempted to solve this from a limited knowledge in sequences and have found out various kinds of sequences do not work but fail to see what it is that could make it work.

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    $\begingroup$ Seems slightly (at least) related to this: encyclopediaofmath.org/index.php/Additive_basis But I think you know already about additive bases. $\endgroup$ – user757601 Mar 25 at 0:37
  • $\begingroup$ See math.stackexchange.com/questions/3579462. (Not an exact duplicate because it doesn't ask for a general test to determine whether a sequence is an alt-basis.) What is the source of this question? $\endgroup$ – joriki Mar 25 at 6:47
  • $\begingroup$ Do you know of any alt-bases that are not of the form $\{b_1,b_2,\dots,b_k,a_{k+1},a_{k+2},\dots\}$ where $A=\{a_1,a_2,\dots\}=\{2^n-1\}$ and $\{b_1,\dots,b_k\}$ is some finite increasing set of integers? $\endgroup$ – Vepir Mar 25 at 21:54
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    $\begingroup$ For example I believe that $\{5\cdot2^{n - 2} + (-1)^{n + 1} 2^{n - 2} - 1\}$ is an alt-basis. $\endgroup$ – Vepir Mar 25 at 22:50

I can’t answer the question, but I can at least give you a systematic large family of alt-bases.

If $A$ is a finite set of positive integers, let $S(A)$ be the set of $A$-expressible integers, and let $S^+(A)$ be the set of $A$-expressible positive integers. Then

$$S(A)=S^+(A)\cup\{-a:a\in S^+(A)\},$$

and if $b>\max A$, then

$$S^+\left(A\cup\{b\}\right)=S^+(A)\cup\{b-s:s\in S^+(A)\}\cup\{b\}.$$

Thus, if $|A|=n$, the maximum number of $A$-expressible positive integers is $2^n-1$, and $\max S(A)=\max A$.

Now suppose that $A=\{a_n:n\in\Bbb Z^+\}$, where $a_n<a_{n+1}$ for each $n\in\Bbb Z^+$. For $n\in\Bbb Z^+$ let $A_n=\{a_k\in A:1\le k\le n\}$. Then each $m\in S(A)$ is uniquely $A$-expressible iff $|S^+(A_n)|=2^n-1$ for each $n\in\Bbb Z^+$. Moreover, $S^+(A)=\Bbb Z^+$ iff for each $k\in S^+(A)$ there is a minimal $n(k)\in\Bbb Z^+$ such that $k\in S^+(A_{n(k)})$. Note that either $n(k)=1$, or $k\in S^+(A_{n(k)})\setminus S^+(A_{n(k)-1})=\{a_{n(k)}-s:s\in S^+(A_{n(k)-1})\}$.

For $n\in\Bbb Z^+$ let

$$a_n=2^n-1=\underbrace{1\ldots 1}_n\text{ in binary},$$

and let $A=\{a_n:n\in\Bbb Z^+\}$. It’s not hard to see that


for each $n\in\Bbb Z^+$, so $A$ is, as you already observed, an alt-basis. For instance, working in binary, we see that

$$\begin{align*} 22&=10110_{\text{two}}\\ &=11111_{\text{two}}-1111_{\text{two}}+111_{\text{two}}-1_{\text{two}}\\ &=31-15+7-1\\ &=a_5-a_4+a_3-a_1. \end{align*}$$

Now let $\ell,m\in\Bbb Z^+$. For $n=1,\ldots,\ell$ let

$$\color{red}{a_n^{(\ell,m)}}=2^ma_n=\underbrace{1\ldots 1}_n\underbrace{0\ldots 0}_m\text{ in binary}.$$

For $n=\ell+k$, where $k=1,\ldots,m$, let

$$\color{blue}{a_n^{(\ell,m)}}=2^{m-k}a_n=\underbrace{1\ldots 1}_n\underbrace{0\ldots 0}_{m-k}\text{ in binary}.$$

Finally, for $n>\ell+m$ let $a_n^{(\ell,m)}=a_n$, and let $A_{(\ell,m)}=\left\{a_n^{(\ell,m)}:n\in\Bbb Z^+\right\}$; then $A_{(\ell,m)}$ is an alt-basis.

For example,

$$\begin{align*} A_{(4,2)}&=\{\color{red}{4},\color{red}{12},\color{red}{28},\color{red}{60},\color{blue}{62},\color{blue}{63},127,\ldots\}\\ &=\{\color{red}{100},\color{red}{1100},\color{red}{11100},\color{red}{111100},\color{blue}{111110},\color{blue}{111111},1111111,\ldots\}\text{ in binary}. \end{align*}$$

To verify this it suffices to show that $S^+\left(\left\{a_n^{(\ell,m)}:1\le n\le \ell+m\right\}\right)=S^+(A_{\ell+m})$. The argument is a bit messy to write out, but the idea is straightforward; I’ll illustrate it with $A_{(4,2)}$. First, it’s clear from the discussion of $A$ that

$$\begin{align*} S^+\left(\{4,12,28,60\}\right)&=S^+\left(4\{1,3,7,15\}\right)\\ &=4S^+\left(\{1,3,7,15\}\right)\\ &=4\{1,2,\ldots,15\}\\ &=\{4,8,12,\ldots,60\}\\ &=4S^+(A_4). \end{align*}$$


$$\begin{align*} S^+(\{4,&12,28,60,62\})=\\ &4S^+(A_4)\cup\left\{|62-s|:s\in S^+(\{4,12,28,60\})\right\}\cup\{62\}=\\ &4S^+(A_4)\cup\left\{|62-s|:s\in\{4,8,12,\ldots,60\}\right\}\cup\{62\}=\\ &4S^+(A_4)\cup\{2,6,10,\ldots,58,62\}=\\ &\{2,4,6,8,\ldots,60,62\}=\\ &2S^+(A_5), \end{align*}$$

and a similar calculation shows that $S^+(\{4,12,28,60,62,63\})=S^+(A_6)$.

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I did not collect the set of all alt-bases, but I did find some useful observations, including:

Alt-basis must contain an infinite number of terms of form $a_k=2^{k}-1,k\in N\subseteq\mathbb N$.

The converse does not hold. At the end, I give examples of alt-bases and not-alt-bases in this context.

Do correct me If I missed anything.

Let $A=\{a_1,a_2,\dots\}$ such that $a_1\lt a_2 \lt \dots$ are positive integers.

Definition. For $A$ to be an "alt-basis", we need to have both the "uniqueness" and "completeness". In other words, every number is expressible in exactly one way via alternating summation of subsets of $A$, which are summed in increasing order.

Definition. A finite (sub)sequence $A|_n=\{a_1,\dots,a_n\}$ is an "alt-prefix" if every integer in $[1,2^{n}-1]$ is uniquely expressible via alternating summation of subsets of $A|_n$ when summed in increasing order. The element $a_n$ is called an "anchor element".

Definition. "Anchor sequence" is a set $\mathcal A(A):=\{a_{n_1},a_{n_2},\dots\}$ of all "anchor elements" $a_{n_1},a_{n_2},\dots$

Notice that a set has $2^n$ subsets minus the empty set and that every subset can be rearranged in an increasing order. We want to assign a distinct value to each of those subsets via the alternating summation, to have an alt-basis. The alt-prefix is defined to cover exactly those $2^n-1$ subsets. It follows that:

Lemma. $A$ is an alt-basis $\iff$ $A$ is a union of alt-prefixes $A=A|_{n_1}\cup A|_{n_2}\cup \dots$

That is, $A$ is an alt-basis if and only if there exists a corresponding infinite anchor sequence $\mathcal A(A)$.

We add two more definitions to write all of this more easily:

Definition. Let $s(\{b_1,\dots,b_n\})$ be the result of the alternating summation of $b_1\lt b_2\lt \dots\le b_n$. Let $s_+$ and $s_-$ always start the alternating summation with $+,-$ respectively. Then $s_+=-s_-$. If $n$ is odd then $s=s_+$ and if $n$ is even then $s=s_-$. This guarantees $s\gt 0$ because the largest element $b_n$ will have a positive sign.

Definition. Define "$n$-th partial subset-sum set" of a positive increasing integer sequence $A$ as:

$$\mathcal S_n(A):=\{s(A_i):A_i\in\mathcal P(A|_n)\}$$

Where $\mathcal P(A|_n)$ is the set of all subsets of $A|_n=\{a_1,a_2,\dots,a_n\}$.

The set of all "anchor elements" $\mathcal A(A)=\{a_{n_1},a_{n_2},\dots\}\subseteq A$ satisfies $S_{n_i}=[1,2^{n_i-1}-1]$ for all $n_i$.

Corollary. $A$ is an alt-basis if and only if it "is covered by the anchor sequence": $\max \mathcal A(A)\to \infty$.

Notice that $\max S_n = a_n$. If $a_n$ is an anchor element, then $\max S_n = 2^n-1$. This gives:

Proposition. If $a_n$ is an anchor element, then $a_n=2^n-1$.

The converse does not hold. For example, in $\{1,4,7\}$ the $a_3=7=2^3-1$ but $a_3$ is not an anchor element, because $S_3=\{1,3,4,6,7\}\ne[1,7]$.

Example $1$. It is not hard to see that $\mathcal A(\{2^n-1\})=\{2^n-1\}$. This is because:

  • $S_1=\{(+1)\}$ $\implies$ $a_1$ is an anchor element.

  • $S_2=\{(+1),(-1+3),(3)\}$ $\implies$ $a_2$ is an anchor element.

  • $S_3=\{(+1),(-1+3),(3),(-3+7),(+1-3+7),(-1+7),(7)\}$ $\implies$ $a_3$ is an anchor element.

  • $\dots$ proceed via induction to show every $a_n$ is an anchor element.

Since $\mathcal A(\{2^n-1\})$ exists and covers the entire $\{2^n-1\}$, the $\{2^n-1\}$ is an alt-basis.

Example $2$. The $\mathcal A(\{n\})=\{1\}$ does not cover the entire $\{n\}$, hence $\{n\}$ is not an alt-basis.

It is not hard to see that $\max S_n = n \lt 2^n-1\implies a_n$ is not an anchor element, for every $n\gt 1$.

Example $3.$ We construct an alt-basis where every $2$nd element is an anchor element.

$$A=\begin{cases} 2^n-1, & n\text{ is even} \\ 2^n+2^{n-1}-1, & n\text{ is odd} \end{cases}$$

Use an inductive argument. Assume $n=2k$, $a_{n}=2^{n}-1$ is an anchor element, which means we have uniquely constructed all $I_0=[1,2^n-1]=S_{n}$ elements. Now we can subtract numbers in this interval from $a_{n+1}$ to see that:

  • $a_{n+1}=2^{n+1}+2^{n}-1$ will cover $I_1=[a_{n+1}-a_{n}, a_{n+1}]=[2^{n+1},2^{n+1}+2^{n}-1]$

Here we see that $I_0\cup I_1 \ne [1,2^{n+1}-1]$ $\implies$ $a_{n+1}$ is not an anchor elemetn.

To see that $a_{n+2}=2^{n+2}-1$ is an anchor element, lets see what will we cover with it:

  • $a_{n+2}$ combined with $I_0$ will cover $I_2=[a_{n+2}-a_{n}, a_{n+2}]=[2^{n+1}+2^{n},2^{n+2}-1]$

  • $a_{n+2}$ combined with $I_1$ will cover $I_3=[a_{n+2}-a_{n+1},a_{n+2}-2^{n+1}]=[2^n,2^{n+1}-1]$

Now observe $I=I_0\cup I_3\cup I_1\cup I_2$ is equal to:

$$ I=[1,2^n-1]\cup[2^n,2^{n+1}-1]\cup[2^{n+1},2^{n+1}+2^{n}-1]\cup[2^{n+1}+2^{n},2^{n+2}-1]=[1,2^{n+2}-1] $$

Implying $a_{n+2}$ covers $I=[1,2^{n+2}-1]=S_{n+2}$, $\implies$ $a_{n+2}$ is an anchor.

It is not hard to check base cases $n=1,2$, and we are done. We have:

$$\mathcal A\left(\left.\begin{cases} 2^n-1, & n\text{ is even} \\ 2^n+2^{n-1}-1, & n\text{ is odd} \end{cases}\right\}\right)=\{2^{2n}-1\}$$

So we have an alt-basis $A$.

Example $4$. It is not hard to show that:

$$ A=\{2^k,2^k+1,2^k+3,2^k+7,\dots,2^k+2^{k}-1,2^{k+2}-1,2^{k+3}-1,\dots\} $$

Is an alt-basis for every $k=0,1,2,\dots$, whose anchors are all elements $a_n,n\gt k$.

Example $5$. The sequence of natural, triangular, tetrahedral,... numbers, or in general, any diagonal of the pascals triangle, is not an alt basis.

This is because for every fixed $d$, there exists $n_0$, such that for all $n\ge n_0$, we have $\binom{n+d-1}{d}<2^n$ implying that $\max S_n\lt 2^n-1$ for all $n\ge n_0$. This implies the sequence of anchors has at most $n_0$ elements, implying $\max\mathcal A(A)\lt \infty$, hence we do not have an alt-basis becuase of inevetable duplicates.

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  • $\begingroup$ I think => part of the initial lemme requires proof. $\endgroup$ – balcinus Mar 31 at 10:07

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