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LSpice
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Is the consecutive sum set large onin general?

It$\DeclareMathOperator\CSS{CSS}$It is well known that for a set $A$ of integers, if $gcd(A) = d$$\gcd(A) = d$, then the set of (integer) linear combinationcombinations of $A$ is $d\mathbb{Z}$. I'm looking for a probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$$\gcd(A) = d$. Let $N$ be large (depending on $A,\varepsilon$$A$, $\varepsilon$) and $\alpha\in A^N$ such that, the density of every $a\in A$ in $\alpha$ satisfies $|\alpha^{-1}(a)|/N\geq \varepsilon$$\lvert\alpha^{-1}(a)\rvert/N\geq \varepsilon$. Let $CCS(\alpha)$$\CSS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$$n$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$$m$, $b = \alpha(n)+\alpha(n+1)+\dotsb+\alpha(n+m-1)$.

Question: Do we have, $|CCS(\alpha) |/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.$\lvert\CSS(\alpha) \rvert/\sum_n\alpha(n)\geq (1-\varepsilon)/d$?

Is consecutive sum set large on general

It is well known that for a set $A$ of integers, if $gcd(A) = d$, then the (integer) linear combination of $A$ is $d\mathbb{Z}$. I'm looking for a probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$. Let $N$ be large (depending on $A,\varepsilon$) and $\alpha\in A^N$ such that, the density of every $a\in A$ in $\alpha$ satisfies $|\alpha^{-1}(a)|/N\geq \varepsilon$. Let $CCS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$.

Question: Do we have, $|CCS(\alpha) |/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.

Is the consecutive sum set large in general?

$\DeclareMathOperator\CSS{CSS}$It is well known that for a set $A$ of integers, if $\gcd(A) = d$, then the set of (integer) linear combinations of $A$ is $d\mathbb{Z}$. I'm looking for a probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $\gcd(A) = d$. Let $N$ be large (depending on $A$, $\varepsilon$) and $\alpha\in A^N$ such that the density of every $a\in A$ in $\alpha$ satisfies $\lvert\alpha^{-1}(a)\rvert/N\geq \varepsilon$. Let $\CSS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n$, $m$, $b = \alpha(n)+\alpha(n+1)+\dotsb+\alpha(n+m-1)$.

Question: Do we have $\lvert\CSS(\alpha) \rvert/\sum_n\alpha(n)\geq (1-\varepsilon)/d$?

deleted 47 characters in body
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Jiayi Liu
Jiayi Liu
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It is well known that for a set $A$ of integers, if $gcd(A) = 1$$gcd(A) = d$, then the (integer) linear combination of $A$ is $\mathbb{Z}$$d\mathbb{Z}$. I'm looking for thea probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$. Let $N$ be large (depending on $A,\varepsilon$) and $\alpha\in \mathbb{N}^N$$\alpha\in A^N$ such that, the density of every $a\in A$ in $\alpha$ satisfies $|\alpha^{-1}(a)/N|\geq \varepsilon$; and $N/\sum_n\alpha(n)\geq \varepsilon$$|\alpha^{-1}(a)|/N\geq \varepsilon$. Let $CCS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$.

Question: Do we have: $|CCS(\alpha)|/\sum_n\alpha(n)\geq (1-\varepsilon)/d$, $|CCS(\alpha) |/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.

It is well known that for a set $A$ of integers, if $gcd(A) = 1$, then the (integer) linear combination of $A$ is $\mathbb{Z}$. I'm looking for the probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$. Let $N$ be large (depending on $A,\varepsilon$) and $\alpha\in \mathbb{N}^N$ such that, the density of every $a\in A$ in $\alpha$ satisfies $|\alpha^{-1}(a)/N|\geq \varepsilon$; and $N/\sum_n\alpha(n)\geq \varepsilon$. Let $CCS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$.

Question: Do we have: $|CCS(\alpha)|/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.

It is well known that for a set $A$ of integers, if $gcd(A) = d$, then the (integer) linear combination of $A$ is $d\mathbb{Z}$. I'm looking for a probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$. Let $N$ be large (depending on $A,\varepsilon$) and $\alpha\in A^N$ such that, the density of every $a\in A$ in $\alpha$ satisfies $|\alpha^{-1}(a)|/N\geq \varepsilon$. Let $CCS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$.

Question: Do we have, $|CCS(\alpha) |/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.

Post Undeleted by Jiayi Liu
Post Deleted by Jiayi Liu
Source Link
Jiayi Liu
Jiayi Liu
  • 909
  • 4
  • 10

Is consecutive sum set large on general

It is well known that for a set $A$ of integers, if $gcd(A) = 1$, then the (integer) linear combination of $A$ is $\mathbb{Z}$. I'm looking for the probability generalization of this, namely the following.

Let $\varepsilon>0$, a finite set $A$ of positive integers with $gcd(A) = d$. Let $N$ be large (depending on $A,\varepsilon$) and $\alpha\in \mathbb{N}^N$ such that, the density of every $a\in A$ in $\alpha$ satisfies $|\alpha^{-1}(a)/N|\geq \varepsilon$; and $N/\sum_n\alpha(n)\geq \varepsilon$. Let $CCS(\alpha)$ (consecutive sum set) denote the set of $b\in\mathbb{N}$ such that for some $n,m$, $b = \alpha(n)+\alpha(n+1)+\cdots+\alpha(n+m-1)$.

Question: Do we have: $|CCS(\alpha)|/\sum_n\alpha(n)\geq (1-\varepsilon)/d$.