Given $a\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+aU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap a(U-U)=\{0\}$ (or equivalently: $u+av=u'+av' \implies u=u', v=v'$)?

Here is an example of near miss: if $a=10$ define $U$ as the set of positive reals with $0$ digits in odd places. Then the decomposition fails to be unique, but only for finite decimals, such as $1=1+10\times0=0.0909... +10\times 0.0909...$

This question is a simple case of a larger, still fuzzy problem I'm thinking about, but I don't want to jump into that without knowing what surely must already have been studied in depth.

I welcome suggestions or edits for better title and tags.

Given $b\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+bU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap b(U-U)=\{0\}$ (or equivalently: $u+bv=u'+bv' \implies u=u', v=v'$)?

Here is an example of near miss: if $b=10$ define $U$ as the set of positive reals with $0$ digits in odd places. Then the decomposition fails to be unique, but only for finite decimals, such as $1=1+10\times0=0.0909... +10\times 0.0909...$

This question is a simple case of a larger, still fuzzy problem I'm thinking about, but I don't want to jump into that without knowing what surely must already have been studied in depth.

I welcome suggestions or edits for better title and tags.

Given $a\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+aU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap a(U-U)=\{0\}$ (or equivalently: $u+xv=u'+av' \implies u=u', v=v'$)?

Here is an example of near miss: if $a=10$ define $U$ as the set of positive reals with $0$ digits in odd places. Then the decomposition fails to be unique, but only for finite decimals, such as $1=1+10\times0=0.0909... +10\times 0.0909...$

This question is a simple case of a larger, still fuzzy problem I'm thinking about, but I don't want to jump into that without knowing what surely must already have been studied in depth.

I welcome suggestions or edits for better title and tags.

Given $a\in \mathbb{R}_{>1}$ is there $U\subseteq\mathbb{R}_{\ge 0}$ such that $U+aU=\mathbb{R}_{\ge 0}$ and $(U-U)\cap a(U-U)=\{0\}$ (or equivalently: $u+av=u'+av' \implies u=u', v=v'$)?

Here is an example of near miss: if $a=10$ define $U$ as the set of positive reals with $0$ digits in odd places. Then the decomposition fails to be unique, but only for finite decimals, such as $1=1+10\times0=0.0909... +10\times 0.0909...$

This question is a simple case of a larger, still fuzzy problem I'm thinking about, but I don't want to jump into that without knowing what surely must already have been studied in depth.

I welcome suggestions or edits for better title and tags.