Subexponentially increasing sums of partial quotients in continued fraction expansions

We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S\_n(x)=\sum\_{j=1}^n a\_j(x)$, where $x=[a\_1(x), a\_2(x), \cdots ]$ is the continued fraction expansion of an irrational $x\in (0,1)$. Precisely, for an increasing function $\varphi: \mathbb{N} \rightarrow \mathbb{N}$, one is interested in the Hausdorff dimension of the sets$$E\_\varphi = \left\{x\in (0,1): \lim\_{n\to\infty} \frac {S\_n(x)} {\varphi(n)} =1\right\}.$$Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case $\exp(n^\gamma), \ \gamma \in [1/2, 1)$. We show that when $\gamma \in [1/2, 1)$, $E\_\varphi$ has Hausdorff dimension $1/2$. Thus, surprisingly, the dimension has a jump from $1$ to $1/2$ at $\varphi(n)=\exp(n^{1/2})$. In a similar way, the distribution of the largest partial quotient is also studied.Comment: 12 pages. More details for the proof of Theorem 1.2. are adde

On two subgroups of U(n), important for quantum computing

ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n X n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n X n unitary diagonal matrices with first entry equal to 1

SU(3)X SU(2)XU(1) Chiral Models from Intersecting D4-/D5-branes

We clarify RR tadpole cancellation conditions for intersecting D4-/D5-branes. We find all of the D4-brane models which have D=4 three-generation chiral fermions with the SU(3)XSU(2)XU(1)^n symmetries. For the D5-brane case, we present a solution to the conditions which gives exactly the matter contents of standard model with U(1) anomalies.Comment: 6 pages, submitted to Progress Letter