A lattice walk with all steps having the same length $d$ is called a $d$-walk. Denote by ${\mathcal{T}}_{d}$ the terminal set, that is, the set of all lattice points that can be reached from the origin by means of a $d$-walk. We examine some geometric and algebraic properties of the terminal set. After observing that $({\mathcal{T}}_{d},+)$ is a normal subgroup of the group $(\mathbb{Z}^{N},+)$, we ask questions about the quotient group $\mathbb{Z}^{N}/{\mathcal{T}}_{d}$ and give the number of elements of $\mathbb{Z}^{2}/{\mathcal{T}}_{d}$ in terms of $d$. To establish this result, we use several consequences of Fermat’s theorem about representations of prime numbers of the form $4k+1$ as the sum of two squares. One of the consequences is the fact, observed by Sierpiński, that every natural power of such a prime number has exactly one relatively prime representation. We provide explicit formulas for the relatively prime integers in this representation.

In this paper the authors prove the following result.

Let $\alpha$ be an irrational number. Then for any $\varepsilon > 0$, there are infinitely many prime numbers p such that $\| \alpha p \| < p^{-16/49 +\varepsilon}$.

The exponent $\frac{16}{49}$ improves on $\frac{9}{28}$, which was obtained recently by the second author [Sci. China Ser. A 43 (2000) 703-721]. The result is very close to the exponent $\frac{1}{3}$, which can be obtained under the Generalized Riemann Hypothesis.

Previous approaches to this problem have all used the same basic estimates for the trigonometric sums that arise. However, the present proof uses new bounds, which depend on the Kloosterman sum and also on a counting problem in the geometry of numbers. In addition new techniques for the sieve method are applied. The most significant feature of the new approach is that, unlike previous methods, the exponential sum estimates remain non-trivial for the exponent $\frac{1}{3}$. This gives one hope for an unconditional result as good as that available under the Generalized Riemann Hypothesis.

2000 Mathematical Subject Classification: 11N36, 11J71.