Infinite binary words containing repetitions of odd period

•∄ infinite 3+3+-free binary word avoiding all squares of odd periods.•∄ infinite binary word, simultaneously avoiding cubes and squares of even periods.•∃ an infinite 3+3+-free binary word avoiding squares of even periods.•∃ an infinite 3+3+-free binary word avoiding squares of period >3.•∃ an infinite 3+3+-free binary word with at most 1 cube and 7 squares of odd periods.

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is its length divided by its smallest period. In this article, we give some new results on the trade-off between the number of squares and the number of cubes in infinite binary words whose square factors have odd periods.