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We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete. [ABSTRACT FROM AUTHOR]

The (bitwise) complement x of a binary word x is obtained by changing each 0 in x to 1 and vice versa. An antisquare is a nonempty word of the form x x. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is (5 +√5)/2. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is good if the only antisquares it contains are 01 and 10. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length n and determine the repetition threshold between polynomial and exponential growth for the number of good words. [ABSTRACT FROM AUTHOR]