How close can we get to linear multiply, add, and compare (on integers)?
Probably not the best answer, but perhaps this is a useful starting point. If we wish to represent a non-negative integer, we can store it as a set of residues modulo sequential prime numbers starting from 2. In this form comparison is potentially hard, but multiplication and addition can be done pretty quickly. The product of the first $n$ primes is approximated by $$p_n\# =\exp((1+\mathcal{o}(1) )n\log n).$$
Hence to represent an integer $N$ you require residues modulo the first $n$ primes, where $N&lt;\exp((1+\mathcal{o}(1) )n\log n)$. Since we can represent any $N&lt;p_n\#$ using residues modulo the first $n$ prime residues, we can take $n \log n \propto \log(N)$. Addition and multiplication can be done directly between pairs of residues. There are $n$ such pairs, with the maximum prime being around $n\log(n)$. Thus addition should be in $O(n\log\log(N))$, while multiplication via Schonhage-Strassen should be in $O(n\log\log(N)\log\log\log\log(N)\log\log\log(N))$. Since $n \log n \propto \log N$, then a rough approximation gives $n$ in $O(\log N/\log\log N)$. This would give the complexity of addition and multiplication as approximately $O(\log N)$ and $O(\log N \log\log\log N \log\log\log\log N)$.