Historical Records of Australian Science publishes peer-reviewed articles and book reviews on the history of science and scientists in Australia and the southwest Pacific, biographical memoirs of deceased Fellows of the Academy, and an annual bibliography of the history of Australian science

An investigation of the size of $S+S$ for a finite Beatty sequence $S=(s_i)=(\lfloor i\alpha+\gamma \rfloor)$, where $\lfloor \hphantom{x} \rfloor$ denotes "floor", $\alpha$, $\gamma$ are real with $\alpha\ge 1$, and $0\le i \le k-1$ and $k\ge 3$. For $\alpha>2$, it is shown that $|S+S|$ depends on the number of "centres" of the Sturmian word $\Delta S=(s_i-s_{i-1})$, and hence that $3(k-1)\le |S+S|\le 4k-6$ if $S$ is not an arithmetic progression. A formula is obtained for the number of centres of certain finite periodic Sturmian words, and this leads to further information about $|S+S|$ in terms of finite nearest integer continued fractions.

An asymptotic estimate is obtained for the number of partitions of the positive integer n into distinct parts, each of which is at least m. The estimate holds uniformly with respect to positive m such that m = 0(n(log n)-9), as n --> infinity.

Estimates are given for the number of variables required to solve simultaneous diagonal (or additive) congruences, with applications to p-adic equations and equations over GF(p). The main tool is a specialisation of a result on partitioning matroids.

AbstractAn approach to p-adic interpolation via divided differences is used to give alternative proofs of results of van der Poorten on p-adic exponential polynomials and to derive a p-adic analogue of Turan's first main theorem on sums of powers.

An investigation of bounds in terms of λ 1 , ... , λ 9 for the least non-trivial solution of the Diophantine equation λ 1 x 1 3 + ... + λ 9 x 9 3 = 0, together with an investigation of the corresponding problem for a diagonal cubic Diophantine inequality.

An investigation of the size of S + S for a nite Beatty sequence S =( si )= (bi + c), where bc denotes \floor", , are real with 1, and 0 i k 1 and k 3. For >2, it is shown that jS +Sj depends on the number of \centres" of the Sturmian word S =( si si 1), and hence that 3(k 1) jS +S j4k 6i fS is not an arithmetic progression. A formula is obtained for the number of centres of certain nite periodic Sturmian words, and this leads to further information about jS + Sj in terms of nite nearest integer continued fractions.