Throughout this paper a ring will mean a commutative ring with identity element. If A is an ideal of the ring R and P is a minimal prime ideal of A, then the intersection Q of all P-primary ideals which contain A is called the isolated primary component of A belonging to P. The ideal Q can also be described as the set of all elements x∈R such that xr∈A for some r∈R\P. If {Pα} is the collection of all minimal prime ideals of A and Qα is the isolated primary component of A belonging to Pα, then is called the kernel of A.

Abstract. Let Φ be in the disc algebra H∞ ∩ C(T) with its restriction to T in the Zygmund space of smooth functions λ*(T). If P(Φ') ⊂ T is the set of Plessner points of Φ' and if F = Φ + Ψ, where Ψ∈C1(T), it is shown that F(P(Φ')) ⊆ C is a set of zero linear Hausdorff measure. Applications are given to the spectral theory of multiplication operators.

The following problem was proposed by Professor N. J. Fine: to prove that there do not exist rational functions F1, F2, F3 of x1, x2, x3, x4, with real coefficients, such that

identically. In the present note I give a proof.†

Let $K$ be an algebraic number field of degree $d\geqslant 3$, $\unicode[STIX]{x1D70E}_{1},\unicode[STIX]{x1D70E}_{2},\ldots ,\unicode[STIX]{x1D70E}_{d}$ the embeddings of $K$ into $\mathbb{C}$, $\unicode[STIX]{x1D6FC}$ a non-zero element in $K$, $a_{0}\in \mathbb{Z}$, $a_{0}>0$ and

$$\begin{eqnarray}F_{0}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC})Y).\end{eqnarray}$$

$\unicode[STIX]{x1D710}$

$K$

$a\in \mathbb{Z}$

$F_{0}(X,Y)\in \mathbb{Z}[X,Y]$

$\unicode[STIX]{x1D710}^{a}$

$a\in \mathbb{Z}$

$\unicode[STIX]{x1D710}$

$$\begin{eqnarray}F_{a}(X,Y)=a_{0}\mathop{\prod }_{i=1}^{d}(X-\unicode[STIX]{x1D70E}_{i}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})Y).\end{eqnarray}$$

$m>0$

$(x,y,a)\in \mathbb{Z}^{3}$

$$\begin{eqnarray}0<|F_{a}(x,y)|\leqslant m\end{eqnarray}$$

$xy\not =0$

$\mathbb{Q}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D710}^{a})=K$

$m$

$K$

$F_{0}$

$\unicode[STIX]{x1D710}$

$d$

In this note we generalize the following result of Sawyer [5]:

Theorem 1. There is a function ψ on ℝ such that, whenever g is a real-valued Borel measurable function on (a subset of) ℝ. × ℝn-1 with the property that y ↦ g(y, t) is C1 for a.e. t, the set

has measure zero.

In this paper we study the Duffin–Schaeffer conjecture, which claims that $\unicode[STIX]{x1D706}(\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n})=1$ if and only if $\sum _{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n})=\infty$ , where $\unicode[STIX]{x1D706}$ denotes the Lebesgue measure on $\mathbb{R}/\mathbb{Z}$ ,

$$\begin{eqnarray}{\mathcal{E}}_{n}={\mathcal{E}}_{n}(\unicode[STIX]{x1D713})=\mathop{\bigcup }_{\substack{ m=1 \\ (m,n)=1}}^{n}\bigg(\frac{m-\unicode[STIX]{x1D713}(n)}{n},\frac{m+\unicode[STIX]{x1D713}(n)}{n}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D713}$

$\bigcap _{m=1}^{\infty }\bigcup _{n=m}^{\infty }{\mathcal{E}}_{n}$

$\bigcup _{n=1}^{\infty }{\mathcal{E}}_{n}$

$C>0$

$$\begin{eqnarray}\unicode[STIX]{x1D706}\bigg(\mathop{\bigcup }_{n=1}^{\infty }{\mathcal{E}}_{n}\bigg)\geqslant C\min \bigg\{\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}({\mathcal{E}}_{n}),1\bigg\}.\end{eqnarray}$$

$p$

In this paper we discuss some dimension results for triangle sets of compact sets in $\mathbb{R}^{2}$. In particular we prove that for any compact set $F$ in $\mathbb{R}^{2}$, the triangle set $\unicode[STIX]{x1D6E5}(F)$ satisfies

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant {\textstyle \frac{3}{2}}\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>1$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant 1+\dim _{\text{A}}F.\end{eqnarray}$$

$\dim _{\text{A}}F>4/3$

$$\begin{eqnarray}\dim _{\text{A}}\unicode[STIX]{x1D6E5}(F)\geqslant \min \{{\textstyle \frac{5}{2}}\dim _{\text{A}}F-1,3\}.\end{eqnarray}$$

$F$

$$\begin{eqnarray}\text{}\underline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\text{}\underline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F)\quad \text{and}\quad \overline{\dim _{\text{B}}}F\geqslant {\textstyle \frac{3}{2}}\overline{\dim _{\text{B}}}\unicode[STIX]{x1D6E5}(F).\end{eqnarray}$$

The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

Let where a0≠0, m≥2, n = e1 + … + em and the ξi(1≤i≤m) are the distinct zeros of f in some algebraic closure of the p-adic field . Then K = (ξ1, ξ2, …, ξm) is a finite separable extension of and we denote by “ord” the unique extension of the (additive) p-adic valuation on to K, normalized with ord p = 1.

The lower dimensional Busemann-Petty problem asks whether origin-symmetric convex bodies in ℝ n with smaller i-dimensional central sections necessarily have smaller volume. A generalization of this problem is studied, when the volumes are measured with weights satisfying certain conditions. The case of hyperplane sections (i = n − 1) has been studied by A. Zvavitch.