Additivity of the ideal of microscopic sets Adam Kwela University of Gdańsk, Poland 20th June 2016 SETTOP 2016, Novi Sad, Serbia Adam Kwela Additivity of the ideal of microscopic sets 1/10
Definition (J. Appell) A set M ⊆ R is called microscopic (M ∈ M ) if for all ε ∈ ( 0 , 1 ) there is a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � ε k for all k ∈ N . Fact M is a σ -ideal. Question (G. Horbaczewska) Is add ( M ) = 2 ω under Martin’s axiom? add ( I ) = min � ∈ I � card ( A ) : A ⊆ I ∧ � A / Fact 2 ω = non ( M ) = cov ( M ) = cof ( M ) under Martin’s axiom. Adam Kwela Additivity of the ideal of microscopic sets 2/10
Definition (J. Appell) A set M ⊆ R is called microscopic (M ∈ M ) if for all ε ∈ ( 0 , 1 ) there is a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � ε k for all k ∈ N . Fact M is a σ -ideal. Question (G. Horbaczewska) Is add ( M ) = 2 ω under Martin’s axiom? add ( I ) = min � ∈ I � card ( A ) : A ⊆ I ∧ � A / Fact 2 ω = non ( M ) = cov ( M ) = cof ( M ) under Martin’s axiom. Adam Kwela Additivity of the ideal of microscopic sets 2/10
Definition (J. Appell) A set M ⊆ R is called microscopic (M ∈ M ) if for all ε ∈ ( 0 , 1 ) there is a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � ε k for all k ∈ N . Fact M is a σ -ideal. Question (G. Horbaczewska) Is add ( M ) = 2 ω under Martin’s axiom? add ( I ) = min � ∈ I � card ( A ) : A ⊆ I ∧ � A / Fact 2 ω = non ( M ) = cov ( M ) = cof ( M ) under Martin’s axiom. Adam Kwela Additivity of the ideal of microscopic sets 2/10
Definition (J. Appell) A set M ⊆ R is called microscopic (M ∈ M ) if for all ε ∈ ( 0 , 1 ) there is a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � ε k for all k ∈ N . Fact M is a σ -ideal. Question (G. Horbaczewska) Is add ( M ) = 2 ω under Martin’s axiom? add ( I ) = min � ∈ I � card ( A ) : A ⊆ I ∧ � A / Fact 2 ω = non ( M ) = cov ( M ) = cof ( M ) under Martin’s axiom. Adam Kwela Additivity of the ideal of microscopic sets 2/10
Let ( f n ) n be a sequence of increasing functions f n : ( 0 , 1 ) → ( 0 , 1 ) such that the following definition makes sense (i.e., lim x → 0 + f n ( x ) = 0 for all n and there is x 0 ∈ ( 0 , 1 ) such that for all 0 < x < x 0 the sequence ( f n ( x )) n is non-increasing and � n f n ( x ) < + ∞ ). Definition (G. Horbaczewska) A set M ⊆ R is called ( f n ) -microscopic (M ∈ M ( f n ) ) if for all ε ∈ ( 0 , 1 ) there is a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � f k ( ε ) for all k ∈ N . F is the family of all ( f n ) n satisfying the above conditions. Proposition (G. Horbaczewska) � ( f n ) n ∈F M ( f n ) is the family of all sets of strong measure zero. Proposition (Czudek, K., Mrożek, Wołoszyn) � ( f n ) n ∈F M ( f n ) is the family of all Lebesgue null sets. Adam Kwela Additivity of the ideal of microscopic sets 3/10
Let ( f n ) n be a sequence of increasing functions f n : ( 0 , 1 ) → ( 0 , 1 ) such that the following definition makes sense (i.e., lim x → 0 + f n ( x ) = 0 for all n and there is x 0 ∈ ( 0 , 1 ) such that for all 0 < x < x 0 the sequence ( f n ( x )) n is non-increasing and � n f n ( x ) < + ∞ ). Definition (G. Horbaczewska) A set M ⊆ R is called ( f n ) -microscopic (M ∈ M ( f n ) ) if for all ε ∈ ( 0 , 1 ) there is a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � f k ( ε ) for all k ∈ N . F is the family of all ( f n ) n satisfying the above conditions. Proposition (G. Horbaczewska) � ( f n ) n ∈F M ( f n ) is the family of all sets of strong measure zero. Proposition (Czudek, K., Mrożek, Wołoszyn) � ( f n ) n ∈F M ( f n ) is the family of all Lebesgue null sets. Adam Kwela Additivity of the ideal of microscopic sets 3/10
Definition � x 2 n � A set M ⊆ R is called nanoscopic if it is -microscopic. Question (G. Horbaczewska) Is the family of all nanoscopic sets an ideal? Theorem (Czudek, K., Mrożek, Wołoszyn) No! Question How does the ideal/ σ -ideal generated by nanoscopic sets look like? Is it equal to M ( g n ) for some ( g n ) n ∈ F ? Question Let f n ( ε ) = ε 2 n for all ε ∈ ( 0 , 1 ) and n ∈ N . Is M ( f n ) an ideal? Adam Kwela Additivity of the ideal of microscopic sets 4/10
Definition � x 2 n � A set M ⊆ R is called nanoscopic if it is -microscopic. Question (G. Horbaczewska) Is the family of all nanoscopic sets an ideal? Theorem (Czudek, K., Mrożek, Wołoszyn) No! Question How does the ideal/ σ -ideal generated by nanoscopic sets look like? Is it equal to M ( g n ) for some ( g n ) n ∈ F ? Question Let f n ( ε ) = ε 2 n for all ε ∈ ( 0 , 1 ) and n ∈ N . Is M ( f n ) an ideal? Adam Kwela Additivity of the ideal of microscopic sets 4/10
Definition � x 2 n � A set M ⊆ R is called nanoscopic if it is -microscopic. Question (G. Horbaczewska) Is the family of all nanoscopic sets an ideal? Theorem (Czudek, K., Mrożek, Wołoszyn) No! Question How does the ideal/ σ -ideal generated by nanoscopic sets look like? Is it equal to M ( g n ) for some ( g n ) n ∈ F ? Question Let f n ( ε ) = ε 2 n for all ε ∈ ( 0 , 1 ) and n ∈ N . Is M ( f n ) an ideal? Adam Kwela Additivity of the ideal of microscopic sets 4/10
Definition � x 2 n � A set M ⊆ R is called nanoscopic if it is -microscopic. Question (G. Horbaczewska) Is the family of all nanoscopic sets an ideal? Theorem (Czudek, K., Mrożek, Wołoszyn) No! Question How does the ideal/ σ -ideal generated by nanoscopic sets look like? Is it equal to M ( g n ) for some ( g n ) n ∈ F ? Question Let f n ( ε ) = ε 2 n for all ε ∈ ( 0 , 1 ) and n ∈ N . Is M ( f n ) an ideal? Adam Kwela Additivity of the ideal of microscopic sets 4/10
Theorem (Czudek, K., Mrożek, Wołoszyn) If A is nanoscopic and B ∈ SMZ , then A ∪ B is nanoscopic. M ⊆ R is of strong measure zero ( M ∈ SMZ ) if for every sequence ( ε n ) n of positive reals there exists a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � ε k for all k ∈ N . Theorem (Czudek, K., Mrożek, Wołoszyn) � x n ! � There are an -microscopic (picoscopic) set X and a point x ∈ R such that X ∪ { x } is not picoscopic anymore! Adam Kwela Additivity of the ideal of microscopic sets 5/10
Theorem (Czudek, K., Mrożek, Wołoszyn) If A is nanoscopic and B ∈ SMZ , then A ∪ B is nanoscopic. M ⊆ R is of strong measure zero ( M ∈ SMZ ) if for every sequence ( ε n ) n of positive reals there exists a sequence of intervals ( I k ) k such that M ⊆ � k I k and | I k | � ε k for all k ∈ N . Theorem (Czudek, K., Mrożek, Wołoszyn) � x n ! � There are an -microscopic (picoscopic) set X and a point x ∈ R such that X ∪ { x } is not picoscopic anymore! Adam Kwela Additivity of the ideal of microscopic sets 5/10
Theorem (Czudek, K., Mrożek, Wołoszyn) Let ( f n ) n ∈ F . Assume that X ∈ M ( f n ) satisfies at least one of the following conditions: X can be covered by an ( f n ) -microscopic F σ set; X is an unbounded interval; X is bounded. Then X ∪ Y ∈ M ( f n ) for any Y ∈ SMZ . Question Let ( f n ) n ∈ F and X ∈ M ( f n ) be such that I ⊆ X for some interval I. Is X ∪ Y ∈ M ( f n ) for any Y ∈ SMZ ? Adam Kwela Additivity of the ideal of microscopic sets 6/10
Theorem (Czudek, K., Mrożek, Wołoszyn) Let ( f n ) n ∈ F . Assume that X ∈ M ( f n ) satisfies at least one of the following conditions: X can be covered by an ( f n ) -microscopic F σ set; X is an unbounded interval; X is bounded. Then X ∪ Y ∈ M ( f n ) for any Y ∈ SMZ . Question Let ( f n ) n ∈ F and X ∈ M ( f n ) be such that I ⊆ X for some interval I. Is X ∪ Y ∈ M ( f n ) for any Y ∈ SMZ ? Adam Kwela Additivity of the ideal of microscopic sets 6/10
Definition A set is in M ⋆ ( f n ) if it can be covered by an ( f n ) -microscopic F σ set. M ( f n ) \ M ⋆ ( f n ) � = ∅ for any ( f n ) n ∈ F . Theorem (Czudek, K., Mrożek, Wołoszyn) M ⋆ ( f n ) is a σ -ideal for any ( f n ) n ∈ F . Proposition (K.) � � x ln ( n + 1 ) �� = 2 ω . Assume Martin’s axiom. Then add M Adam Kwela Additivity of the ideal of microscopic sets 7/10
Definition A set is in M ⋆ ( f n ) if it can be covered by an ( f n ) -microscopic F σ set. M ( f n ) \ M ⋆ ( f n ) � = ∅ for any ( f n ) n ∈ F . Theorem (Czudek, K., Mrożek, Wołoszyn) M ⋆ ( f n ) is a σ -ideal for any ( f n ) n ∈ F . Proposition (K.) � � x ln ( n + 1 ) �� = 2 ω . Assume Martin’s axiom. Then add M Adam Kwela Additivity of the ideal of microscopic sets 7/10
Definition A set is in M ⋆ ( f n ) if it can be covered by an ( f n ) -microscopic F σ set. M ( f n ) \ M ⋆ ( f n ) � = ∅ for any ( f n ) n ∈ F . Theorem (Czudek, K., Mrożek, Wołoszyn) M ⋆ ( f n ) is a σ -ideal for any ( f n ) n ∈ F . Proposition (K.) � � x ln ( n + 1 ) �� = 2 ω . Assume Martin’s axiom. Then add M Adam Kwela Additivity of the ideal of microscopic sets 7/10
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