multidimensional arithmetic progression

An $n$ -dimensional arithmetic progresssion is a set of the form

	$\\displaystyle Q$	$\\displaystyle=Q(a;q_{1},\\ldots,q_{n};l_{1},\\ldots,l_{n})$
		$\\displaystyle=\\{\\,a+x_{1}q_{1}+\\cdots+x_{n}q_{n}\\mid 0\\leq x_{i}<l_{i}\\text{ %for }i=1,\\ldots,n\\,\\}.$

The length of the progression is defined as $l_{1}\\cdots l_{n}$ . The progression is proper if $|Q|=l_{1}\\cdots l_{n}$ .

References

1 Melvyn B. Nathanson. Additive Number Theory: Inverse Problems and Geometry ofSumsets, volume 165 of GTM. Springer, 1996. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003Zbl 0859.11003.