Normal numbers were introduced by Borel. Normality is certainly a weak notion of randomness; for instance, there are computable numbers which are absolutely normal. In the present paper, we introduce a relativization of normality to a fixed representation system. When we require normality with respect to large sets of such systems, we find variants of normality that imply randomness notions much stronger than absolute normality. The primary classes of numbers investigated in this paper are the supernormal numbers and the highly normal numbers, which we will define. These are relativizations of normality which are robust to all reasonable changes of representation. Among other results, we give a proof that the highly normal numbers are exactly those of computable dimension 1, which we think gives a more natural characterization than was previously known of this interesting class.

翻译：Borel引入了正规数概念。正规性显然是一种较弱的随机性概念；例如，存在绝对正规的可计算数。本文针对固定表示系统引入了正规性的相对化概念。当要求对大量此类系统具有正规性时，我们发现某些正规性变体能够蕴含远强于绝对正规性的随机性概念。本文主要研究超正规数与高正规数这两类数——我们将对此进行定义。这些是正规性的相对化形式，对表示系统中所有合理的变化具有鲁棒性。作为研究成果之一，我们证明了高正规数恰好是那些具有可计算维数1的数，我们认为这为此类有趣的数提供了比先前已知的更自然的刻画。