doi:10.1016/j.tcs.2006.02.022 
Copyright © 2006 Elsevier B.V. All rights reserved.
A note on dimensions of polynomial size circuits
Xiaoyang Gua, 
aDepartment of Computer Science, Iowa State University, Ames, IA 50011, USA
Received 16 May 2005;
revised 8 February 2006;
accepted 16 February 2006.
Communicated by A. Razborov.
Available online 29 March 2006.
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Abstract
In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every i
0, P/poly has ith-order scaled p3-strong dimension 0. We also show that P/polyi.o. has p3-dimension 
and p3-strong dimension 1. Our results improve previous measure results of Lutz [Almost everywhere high nonuniform complexity, J. Comput. Syst. Sci. 44(2) (1992) 220–258] and dimension results of Hitchcock and Vinodchandran [Dimension, entropy rates, and compression, in: Proc. 19th IEEE Conf. Computational Complexity, 2004, pp. 174–183, J. Comput. Syst. Sci., to appear]. Additionally, we establish a Supergale Dilation Theorem, which extends the martingale dilation technique introduced implicitly by Ambos-Spies, Terwijn, and Zheng [Resource bounded randomness and weakly complete problems, Theoret. Comput. Sci. 172(1–2) (1997) 195–207] and made explicit by Juedes and Lutz [Weak completeness in E and E2, Theoret. Comput. Sci. 143(1) (1995) 149–158].
Keywords: Resource-bounded dimension; Resource-bounded measures; Circuit complexity
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[0,1], we are interested in the connectivity properties of the set DIM
