In this paper we provide two equivalent characterizations of the notion of finite-state dimension introduced by Dai, Lathrop, Lutz and Mayordomo (2004). One of them uses Shannon's entropy of nonaligned blocks and generalizes old results of Pillai (1940) and Niven-Zuckerman (1951). The second characterizes finite-state dimension in terms of superadditive functions that satisfy some calibration condition (in particular, superadditive upper bounds for Kolmogorov complexity). The use of superadditive bounds allows us to prove a general sufficient condition for normality that easily implies old results of Champernowne (1933), Besicovitch (1935), Copeland and Erdös (1946), and also a recent result of Calude, Staiger and Stephan (2016).

A major open problem in the area of infinite-duration games is to characterize winning conditions that are determined in finite-memory strategies. Infiniteduration games are usually studied over edge-colored graphs, with winning conditions that are defined in terms of sequences of colors. In this paper, we investigate a restricted class of finite-memory strategies called chromatic finite-memory strategies. While general finite-memory strategies operate with sequences of edges of a game graph, chromatic finite-memory strategies observe only colors of these edges.Recent results in this area show that studying finite-memory determinacy is more tractable when we restrict ourselves to chromatic strategies. On the other hand, as was shown by Le Roux (CiE 2020), determinacy in general finite-memory strategies implies determinacy in chromatic finite-memory strategies. Unfortunately, this result is quite inefficient in terms of the state complexity: to replace a winning strategy with few states of general memory, we might need much more states of chromatic memory. The goal of the present paper is to find out the exact state complexity of this transformation.For every winning condition and for every game graph with n nodes we show the following: if this game graph has a winning strategy with q states of general memory, then it also has a winning strategy with (q + 1) n states of chromatic memory. We also show that this bound is almost tight. For every q and n, we construct a winning condition and a game graph with n + O(1) nodes, where one can win with q states of general memory, but not with q n − 1 states of chromatic memory.

Kopczyński (ICALP 2006) conjectured that prefix-independent half-positional winning conditions are closed under finite unions. We refute this conjecture over finite arenas. For that, we introduce a new class of prefix-independent bi-positional winning conditions called energy conditions over totally ordered groups. We give an example of two such conditions whose union is not half-positional. We also conjecture that every prefix-independent bi-positional winning condition coincides with some energy condition over a totally ordered group on periodic sequences.

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