Lower bounds for the generalized counting function

1999 ◽  
pp. 275-293 ◽  
Author(s):  
Yu. Safarov
Keyword(s):  
Lower Bounds ◽  
Counting Function

scholarly journals ON THE BITS COUNTING FUNCTION OF REAL NUMBERS

2008 ◽  
Vol 85 (1) ◽  
pp. 95-111 ◽  
Author(s):  
TANGUY RIVOAL
Keyword(s):  
Lower Bounds ◽  
Irrational Number ◽  
Counting Function ◽  
Fibonacci Sequence ◽  
Difficult Problem ◽  
Transcendental Number ◽  
Algebraic Numbers ◽  
Real Numbers ◽  
Binary Expansion ◽  
Multiplicative Constant

AbstractLet Bn(x) denote the number of 1’s occurring in the binary expansion of an irrational number x>0. A difficult problem is to provide nontrivial lower bounds for Bn(x) for interesting numbers such as $\sqrt {2}$, e or π: their conjectural simple normality in base 2 is equivalent to Bn(x)∼n/2. In this article, amongst other things, we prove inequalities relating Bn(x+y), Bn(xy) and Bn(1/x) to Bn(x) and Bn(y) for any irrational numbers x,y>0, which we prove to be sharp up to a multiplicative constant. As a by-product, we provide an answer to a question raised by Bailey et al. (D. H. Bailey, J. M. Borwein, R. E. Crandall and C. Pomerance, ‘On the binary expansions of algebraic numbers’, J. Théor. Nombres Bordeaux16(3) (2004), 487–518) concerning the binary digits of the square of a series related to the Fibonacci sequence. We also obtain a slight refinement of the main theorem of the same article, which provides a nontrivial lower bound for Bn(α) for any real irrational algebraic number. We conclude the article with effective or conjectural lower bounds for Bn(x) when x is a transcendental number.

Lower Bounds in Communication Complexity

2007 ◽  
Author(s):  
T. Lee ◽  
A. Shraibman
Keyword(s):  
Lower Bounds ◽  
Communication Complexity

Two Lower Bounds for Shortest Double-Base Number System

2015 ◽  
Vol E98.A (6) ◽  
pp. 1310-1312 ◽  
Author(s):  
Parinya CHALERMSOOK ◽  
Hiroshi IMAI ◽  
Vorapong SUPPAKITPAISARN
Keyword(s):  
Lower Bounds ◽  
Number System ◽  
Base Number ◽  
Double Base

Lower bounds on the dimension of the Rauzy gasket

2020 ◽  
Vol 148 (2) ◽  
pp. 321-327
Author(s):  
Rodolfo Gutiérrez-Romo ◽  
Carlos Matheus
Keyword(s):  
Lower Bounds
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