On the Duffin-Schaeffer conjecture Dimitris Koukoulopoulos 1 joint - PowerPoint PPT Presentation

On the Duffin-Schaeffer conjecture Dimitris Koukoulopoulos 1 joint work with James Maynard 2 1 Université de Montréal 2 University of Oxford Second Symposium in Analytic Number Theory Cetraro, Italy 10 July 2019

The problem Given ψ : N → [ 0 , + ∞ ) and α ∈ [ 0 , 1 ] , solve the inequality � � α − a � � � ψ ( q ) � � with a ∈ Z , q ∈ N ( ∗ ) � � q q

The problem Given ψ : N → [ 0 , + ∞ ) and α ∈ [ 0 , 1 ] , solve the inequality � � α − a � � � ψ ( q ) � � with a ∈ Z , q ∈ N ( ∗ ) � � q q (possibly imposing the coprimality condition gcd ( a , q ) = 1)

The problem Given ψ : N → [ 0 , + ∞ ) and α ∈ [ 0 , 1 ] , solve the inequality � � α − a � � � ψ ( q ) � � with a ∈ Z , q ∈ N ( ∗ ) � � q q (possibly imposing the coprimality condition gcd ( a , q ) = 1) Dirichlet: when ψ ( q ) = 1 / q , then ( ∗ ) has infinitely many solutions for all α ∈ [ 0 , 1 ] .

The problem Given ψ : N → [ 0 , + ∞ ) and α ∈ [ 0 , 1 ] , solve the inequality � � α − a � � � ψ ( q ) � � with a ∈ Z , q ∈ N ( ∗ ) � � q q (possibly imposing the coprimality condition gcd ( a , q ) = 1) Dirichlet: when ψ ( q ) = 1 / q , then ( ∗ ) has infinitely many solutions for all α ∈ [ 0 , 1 ] . Question: can we solve ( ∗ ) if ψ is more irregular?

The problem Given ψ : N → [ 0 , + ∞ ) and α ∈ [ 0 , 1 ] , solve the inequality � α − a � � � � ψ ( q ) � � with a ∈ Z , q ∈ N ( ∗ ) � � q q (possibly imposing the coprimality condition gcd ( a , q ) = 1) Dirichlet: when ψ ( q ) = 1 / q , then ( ∗ ) has infinitely many solutions for all α ∈ [ 0 , 1 ] . Question: can we solve ( ∗ ) if ψ is more irregular? Caveat: There might be exceptional α ’s.

The problem Given ψ : N → [ 0 , + ∞ ) and α ∈ [ 0 , 1 ] , solve the inequality � α − a � � � � ψ ( q ) � � with a ∈ Z , q ∈ N ( ∗ ) � � q q (possibly imposing the coprimality condition gcd ( a , q ) = 1) Dirichlet: when ψ ( q ) = 1 / q , then ( ∗ ) has infinitely many solutions for all α ∈ [ 0 , 1 ] . Question: can we solve ( ∗ ) if ψ is more irregular? Caveat: There might be exceptional α ’s. Goal: understand when set of exceptional α ’s has null measure

Khinchin’s theorem � a q − ψ ( q ) , a q + ψ ( q ) � � K q := q q 0 � a � q

Khinchin’s theorem � a q − ψ ( q ) , a q + ψ ( q ) � � K q := q q 0 � a � q K := lim sup K q q →∞

Khinchin’s theorem � a q − ψ ( q ) , a q + ψ ( q ) � � K q := q q 0 � a � q K := lim sup K q q →∞ = { α ∈ [ 0 , 1 ] : α ∈ K q for infinitely many q }

Khinchin’s theorem � a q − ψ ( q ) , a q + ψ ( q ) � � K q := q q 0 � a � q K := lim sup K q q →∞ = { α ∈ [ 0 , 1 ] : α ∈ K q for infinitely many q } Note that λ ( K q ) ≍ ψ ( q ) ( λ = Lebesgue measure )

Khinchin’s theorem � a q − ψ ( q ) , a q + ψ ( q ) � � K q := q q 0 � a � q K := lim sup K q q →∞ = { α ∈ [ 0 , 1 ] : α ∈ K q for infinitely many q } Note that λ ( K q ) ≍ ψ ( q ) ( λ = Lebesgue measure ) � • ‘easy’ direction of Borel-Cantelli : ψ ( q ) < ∞ ⇒ λ ( K ) = 0 . q

Khinchin’s theorem � a q − ψ ( q ) , a q + ψ ( q ) � � K q := q q 0 � a � q K := lim sup K q q →∞ = { α ∈ [ 0 , 1 ] : α ∈ K q for infinitely many q } Note that λ ( K q ) ≍ ψ ( q ) ( λ = Lebesgue measure ) � • ‘easy’ direction of Borel-Cantelli : ψ ( q ) < ∞ ⇒ λ ( K ) = 0 . q • Khinchin (1924) proved a partial converse: � q ψ ( q ) ց & ψ ( q ) = ∞ ⇒ λ ( K ) = 1 . q

The Duffin-Schaeffer conjecture Study coprime solutions to | α − a / q | � ψ ( q ) / q to avoid over-counting:

The Duffin-Schaeffer conjecture Study coprime solutions to | α − a / q | � ψ ( q ) / q to avoid over-counting: � a q − ψ ( q ) , a q + ψ ( q ) � � A q := , A = lim sup A q q q q →∞ 1 � a � q gcd ( a , q )= 1

The Duffin-Schaeffer conjecture Study coprime solutions to | α − a / q | � ψ ( q ) / q to avoid over-counting: � a q − ψ ( q ) , a q + ψ ( q ) � � A q := , A = lim sup A q q q q →∞ 1 � a � q gcd ( a , q )= 1 • Here λ ( A q ) = ψ ( q ) ϕ ( q ) / q , so the ‘easy’ Borel-Cantelli lemma yields: ψ ( q ) ϕ ( q ) � < ∞ ⇒ λ ( A ) = 0 q q

The Duffin-Schaeffer conjecture Study coprime solutions to | α − a / q | � ψ ( q ) / q to avoid over-counting: � a q − ψ ( q ) , a q + ψ ( q ) � � A q := , A = lim sup A q q q q →∞ 1 � a � q gcd ( a , q )= 1 • Here λ ( A q ) = ψ ( q ) ϕ ( q ) / q , so the ‘easy’ Borel-Cantelli lemma yields: ψ ( q ) ϕ ( q ) � < ∞ ⇒ λ ( A ) = 0 q q • Duffin and Schaeffer (1941) conjecture a strong converse is also true: ψ ( q ) ϕ ( q ) � = ∞ ⇒ λ ( A ) = 1 . q q

Results on DSC (Duffin-Schaeffer Conjecture)

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞ • Erd˝ os (1970) & Vaaler (1978) : DSC is true when ψ ( q ) = O ( 1 / q ) .

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞ • Erd˝ os (1970) & Vaaler (1978) : DSC is true when ψ ( q ) = O ( 1 / q ) . • Pollington-Vaughan (1990) : DSC is true in all dimensions > 1.

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞ • Erd˝ os (1970) & Vaaler (1978) : DSC is true when ψ ( q ) = O ( 1 / q ) . • Pollington-Vaughan (1990) : DSC is true in all dimensions > 1. ψ ( q ) ϕ ( q ) • DSC with ‘extra divergence’, i.e. when � = ∞ : q qL ( q )

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞ • Erd˝ os (1970) & Vaaler (1978) : DSC is true when ψ ( q ) = O ( 1 / q ) . • Pollington-Vaughan (1990) : DSC is true in all dimensions > 1. ψ ( q ) ϕ ( q ) • DSC with ‘extra divergence’, i.e. when � = ∞ : q qL ( q ) Haynes-Pollington-Velani (2012) : L ( q ) = ( q /ψ ( q )) ε .

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞ • Erd˝ os (1970) & Vaaler (1978) : DSC is true when ψ ( q ) = O ( 1 / q ) . • Pollington-Vaughan (1990) : DSC is true in all dimensions > 1. ψ ( q ) ϕ ( q ) • DSC with ‘extra divergence’, i.e. when � = ∞ : q qL ( q ) Haynes-Pollington-Velani (2012) : L ( q ) = ( q /ψ ( q )) ε . Beresnevich-Harman-Haynes-Velani (2013) : L ( q ) = exp { c ( log log q )( log log log q ) } .

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞ • Erd˝ os (1970) & Vaaler (1978) : DSC is true when ψ ( q ) = O ( 1 / q ) . • Pollington-Vaughan (1990) : DSC is true in all dimensions > 1. ψ ( q ) ϕ ( q ) • DSC with ‘extra divergence’, i.e. when � = ∞ : q qL ( q ) Haynes-Pollington-Velani (2012) : L ( q ) = ( q /ψ ( q )) ε . Beresnevich-Harman-Haynes-Velani (2013) : L ( q ) = exp { c ( log log q )( log log log q ) } . Aistleitner-Lachmann-Munsch-Technau-Zafeiropoulos (2018 preprint) : L ( q ) = ( log q ) ε

Results on DSC (Duffin-Schaeffer Conjecture) • Duffin-Schaeffer (1941): DSC is true when ψ is ‘regular’, i.e. when � q � Q ψ ( q ) ϕ ( q ) / q lim sup > 0 . � q � Q ψ ( q ) Q →∞ • Erd˝ os (1970) & Vaaler (1978) : DSC is true when ψ ( q ) = O ( 1 / q ) . • Pollington-Vaughan (1990) : DSC is true in all dimensions > 1. ψ ( q ) ϕ ( q ) • DSC with ‘extra divergence’, i.e. when � = ∞ : q qL ( q ) Haynes-Pollington-Velani (2012) : L ( q ) = ( q /ψ ( q )) ε . Beresnevich-Harman-Haynes-Velani (2013) : L ( q ) = exp { c ( log log q )( log log log q ) } . Aistleitner-Lachmann-Munsch-Technau-Zafeiropoulos (2018 preprint) : L ( q ) = ( log q ) ε Aistleitner (unpublished) : L ( q ) = ( log log q ) ε .