# Volume 28 - 2005

### 1. A lower bound concerning subset sums which do not cover all the residues modulo $p$.

Let $c>\sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$ where the constant implied in the O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$

### 2. Further Variations on the Six Exponentials Theorem.

According to the Six Exponentials Theorem, a $2\times 3$ matrix whose entries $\lambda_{ij}$ ($i=1,2$, $j=1,2,3$) are logarithms of algebraic numbers has rank $2$, as soon as the two rows as well as the three columns are linearly independent over the field $\BbbQ$ of rational numbers. The main result of the present note is that one at least of the three $2\times 2$ determinants, viz. $\lambda_{21}\lambda_{12}-\lambda_{11}\lambda_{22}, \quad \lambda_{22}\lambda_{13}-\lambda_{12}\lambda_{23}, \quad \lambda_{23}\lambda_{11}-\lambda_{13}\lambda_{21}$ is transcendental.

### 3. On an exponential sum involving the Möbius function

In this paper we study the upper bound for the absolute value of the exponential sum related to the Möbius function unconditionally and present some interesting applications also.

### 4. A lower bound concerning subset sums which do not cover all the residues modulo $p$.

Let $c>\sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$ where the constant implied in the O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$