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Let $\mathrm{d}_\ast$ be the lower asymptotic density on $\mathbb N$ (the non-negative integers), that is, the function $$ \mathcal P(\mathbb N) \to [0, 1] \colon A \mapsto \liminf_{n \to \infty} \frac{\bigl|A \cap [\![ 1, n ]\!]\bigr|}{n}, $$ where $\mathcal P(\mathbb N)$ is the power set of $\mathbb N$ and, for all $a, b \in \mathbb Z$, we denote by $[\![a, b ]\!]$ the discrete interval $\{x \in \mathbb Z : a \le x \le b\}$ from $a$ to $b$.

It is (a special case of) a classical theorem by Kneser that the following holds:

If $\mathrm{d}_\ast(2X) < 2 \mathrm{d}_\ast(X)$ for some $X \in \mathcal P(\mathbb N)$, then there exist an integer $m \ge 1$ and a set $K \subseteq [\![0, m-1 ]\!]$ such that $X$ is contained in $K + m \times \mathbb N$ (a finite union of arithmetic progressions with common difference $m$) and the difference $2K \setminus 2X$ is finite.

Here, $2Y$ is the $2$-fold sum $\{y_1 + y_2: y_1, y_2 \in Y\}$ of a set $Y \in \mathcal P(\mathbb N)$, and $m \times \mathbb N$ is the set $\{mk: k \in \mathbb N\}$ of the multiples of $m$ in $\mathbb N$.

The standard reference for the result is the following paper:

M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484.

I have a copy of Kneser's paper, but I find it challenging to read, partly because it is written in German and partly because the terminology and style are difficult for me to follow. I've also looked into (the 1967 edition of) Halberstam and Roth's Sequences: it seems that the book contains a proof (of a more general version of the theorem), but I've been unable to find it. So, I'd like to ask:

Question. Can anyone more familiar than me with Kneser's work either pinpoint the result (and its proof) in Kneser's original paper, or provide a precise reference to a book or an article with a modern account of the proof?

Let $\mathrm{d}_\ast$ be the lower asymptotic density on $\mathbb N$ (the non-negative integers), that is, the function $$ \mathcal P(\mathbb N) \to [0, 1] \colon A \mapsto \liminf_{n \to \infty} \frac{\bigl|A \cap [\![ 1, n ]\!]\bigr|}{n}, $$ where $\mathcal P(\mathbb N)$ is the power set of $\mathbb N$ and, for all $a, b \in \mathbb Z$, we denote by $[\![a, b ]\!]$ the discrete interval $\{x \in \mathbb Z : a \le x \le b\}$ from $a$ to $b$.

It is (a special case of) a classical theorem by Kneser that the following holds:

If $\mathrm{d}_\ast(2X) < 2 \mathrm{d}_\ast(X)$ for some $X \in \mathcal P(\mathbb N)$, then there exist an integer $m \ge 1$ and a set $K \subseteq [\![0, m-1 ]\!]$ such that $X$ is contained in $Y := K + m \times \mathbb N$ (a finite union of arithmetic progressions with common difference $m$) and the difference $2Y \setminus 2X$ is finite.

Here, $2A$ is the $2$-fold sum $\{a+b: a, b \in A\}$ of a set $A \in \mathcal P(\mathbb N)$, and $m \times \mathbb N$ is the set $\{mk: k \in \mathbb N\}$ of multiples of $m$ in $\mathbb N$.

The standard reference for the result is the following paper:

M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484.

I have a copy of Kneser's paper, but I find it challenging to read, partly because it is written in German and partly because the terminology and style are difficult for me to follow. I've also looked into (the 1967 edition of) Halberstam and Roth's Sequences: it seems that the book contains a proof (of a more general version of the theorem), but I've been unable to find it. So, I'd like to ask:

Question. Can anyone more familiar than me with Kneser's work either pinpoint the result (and its proof) in Kneser's original paper, or provide a precise reference to a book or an article with a modern account of the proof?

Let $\mathrm{d}_\ast$ be the lower asymptotic density on $\mathbb N$ (the non-negative integers), that is, the function $$ \mathcal P(\mathbb N) \to \mathcal P(\mathbb N) \colon A \mapsto \liminf_{n \to \infty} \frac{\bigl|A \cap [\![ 1, n ]\!]\bigr|}{n}, $$ where $\mathcal P(\mathbb N)$ is the power set of $\mathbb N$ and, for all $a, b \in \mathbb Z$, we denote by $[\![a, b ]\!]$ the discrete interval $\{x \in \mathbb Z : a \le x \le b\}$ from $a$ to $b$.

It is (a special case of) a classical theorem by Kneser that the following holds:

If $\mathrm{d}_\ast(2X) < 2 \mathrm{d}_\ast(X)$ for some $X \in \mathcal P(\mathbb N)$, then there exist an integer $m \ge 1$ and a set $K \subseteq [\![0, m-1 ]\!]$ such that $X$ is contained in $K + m \times \mathbb N$ (a finite union of arithmetic progressions with common difference $m$) and the difference $2K \setminus 2X$ is finite.

Here, $2Y$ is the $2$-fold sum $\{y_1 + y_2: y_1, y_2 \in Y\}$ of a set $Y \in \mathcal P(\mathbb N)$, and $m \times \mathbb N$ is the set $\{mk: k \in \mathbb N\}$ of the multiples of $m$ in $\mathbb N$.

The standard reference for the result is the following paper:

M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484.

I have a copy of Kneser's paper, but I find it challenging to read, partly because it is written in German and partly because the terminology and style are difficult for me to follow. I've also looked into (the 1967 edition of) Halberstam and Roth's Sequences: it seems that the book contains a proof (of a more general version of the theorem), but I've been unable to find it. So, I'd like to ask:

Question. Can anyone more familiar than me with Kneser's work either pinpoint the result (and its proof) in Kneser's original paper, or provide a precise reference to a book or an article with a modern account of the proof?

Let $\mathrm{d}_\ast$ be the lower asymptotic density on $\mathbb N$ (the non-negative integers), that is, the function $$ \mathcal P(\mathbb N) \to [0, 1] \colon A \mapsto \liminf_{n \to \infty} \frac{\bigl|A \cap [\![ 1, n ]\!]\bigr|}{n}, $$ where $\mathcal P(\mathbb N)$ is the power set of $\mathbb N$ and, for all $a, b \in \mathbb Z$, we denote by $[\![a, b ]\!]$ the discrete interval $\{x \in \mathbb Z : a \le x \le b\}$ from $a$ to $b$.

It is (a special case of) a classical theorem by Kneser that the following holds:

If $\mathrm{d}_\ast(2X) < 2 \mathrm{d}_\ast(X)$ for some $X \in \mathcal P(\mathbb N)$, then there exist an integer $m \ge 1$ and a set $K \subseteq [\![0, m-1 ]\!]$ such that $X$ is contained in $K + m \times \mathbb N$ (a finite union of arithmetic progressions with common difference $m$) and the difference $2K \setminus 2X$ is finite.

Here, $2Y$ is the $2$-fold sum $\{y_1 + y_2: y_1, y_2 \in Y\}$ of a set $Y \in \mathcal P(\mathbb N)$, and $m \times \mathbb N$ is the set $\{mk: k \in \mathbb N\}$ of the multiples of $m$ in $\mathbb N$.

The standard reference for the result is the following paper:

M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484.

I have a copy of Kneser's paper, but I find it challenging to read, partly because it is written in German and partly because the terminology and style are difficult for me to follow. I've also looked into (the 1967 edition of) Halberstam and Roth's Sequences: it seems that the book contains a proof (of a more general version of the theorem), but I've been unable to find it. So, I'd like to ask:

Question. Can anyone more familiar than me with Kneser's work either pinpoint the result (and its proof) in Kneser's original paper, or provide a precise reference to a book or an article with a modern account of the proof?

Let $\mathrm{d}_\ast$ be the lower asymptotic density on $\mathbb N$ (the non-negative integers), that is, the function $$ \mathcal P(\mathbb N) \to \mathcal P(\mathbb N) \colon A \mapsto \liminf_{n \to \infty} \frac{\bigl|A \cap [\![ 1, n ]\!]\bigr|}{n}, $$ where $\mathcal P(\mathbb N)$ is the power set of $\mathbb N$ and, for all $a, b \in \mathbb Z$, we denote by $[\![a, b ]\!]$ the discrete interval $\{x \in \mathbb Z : a \le x \le b\}$ from $a$ to $b$.

It is (a special case of) a classical theorem by Kneser that the following holds:

If $\mathrm{d}_\ast(2X) < 2 \mathrm{d}_\ast(X)$ for some $X \in \mathcal P(\mathbb N)$, then there exist an integer $m \ge 1$ and a set $K \subseteq [\![0, m-1 ]\!]$ such that $A$ is contained in $K + m \times \mathbb N$ (a finite union of arithmetic progressions with common difference $m$) and the difference $2K \setminus 2A$ is finite.

Here, $2Y$ is the $2$-fold sum $\{y_1 + y_2: y_1, y_2 \in Y\}$ of a set $Y \in \mathcal P(\mathbb N)$, and $m \times \mathbb N$ is the set $\{mk: k \in \mathbb N\}$ of the multiples of $m$ in $\mathbb N$.

The standard reference for the result is the following paper:

M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484.

I have a copy of Kneser's paper, but I find it challenging to read, partly because it is written in German and partly because the terminology and style are difficult for me to follow. I've also looked into (the 1967 edition of) Halberstam and Roth's Sequences: it seems that the book contains a proof (of a more general version of the theorem), but I've been unable to find it. So, I'd like to ask the following:

Question. Can anyone more familiar than me with Kneser's work either pinpoint the result (and its proof) in Kneser's original paper, or provide a precise reference to a book or an article with a modern account of the proof?

Let $\mathrm{d}_\ast$ be the lower asymptotic density on $\mathbb N$ (the non-negative integers), that is, the function $$ \mathcal P(\mathbb N) \to \mathcal P(\mathbb N) \colon A \mapsto \liminf_{n \to \infty} \frac{\bigl|A \cap [\![ 1, n ]\!]\bigr|}{n}, $$ where $\mathcal P(\mathbb N)$ is the power set of $\mathbb N$ and, for all $a, b \in \mathbb Z$, we denote by $[\![a, b ]\!]$ the discrete interval $\{x \in \mathbb Z : a \le x \le b\}$ from $a$ to $b$.

It is (a special case of) a classical theorem by Kneser that the following holds:

If $\mathrm{d}_\ast(2X) < 2 \mathrm{d}_\ast(X)$ for some $X \in \mathcal P(\mathbb N)$, then there exist an integer $m \ge 1$ and a set $K \subseteq [\![0, m-1 ]\!]$ such that $X$ is contained in $K + m \times \mathbb N$ (a finite union of arithmetic progressions with common difference $m$) and the difference $2K \setminus 2X$ is finite.

Here, $2Y$ is the $2$-fold sum $\{y_1 + y_2: y_1, y_2 \in Y\}$ of a set $Y \in \mathcal P(\mathbb N)$, and $m \times \mathbb N$ is the set $\{mk: k \in \mathbb N\}$ of the multiples of $m$ in $\mathbb N$.

The standard reference for the result is the following paper:

M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen, Math. Z. 58 (1953), 459–484.

I have a copy of Kneser's paper, but I find it challenging to read, partly because it is written in German and partly because the terminology and style are difficult for me to follow. I've also looked into (the 1967 edition of) Halberstam and Roth's Sequences: it seems that the book contains a proof (of a more general version of the theorem), but I've been unable to find it. So, I'd like to ask:

Question. Can anyone more familiar than me with Kneser's work either pinpoint the result (and its proof) in Kneser's original paper, or provide a precise reference to a book or an article with a modern account of the proof?