The story of squaring the circle M. Laczkovich E otv os - PowerPoint PPT Presentation

The story of squaring the circle M. Laczkovich E¨ otv¨ os University, Budapest Warwick, July 13, 2017

Theorem (G. Vitali 1905) There is no translation invariant measure µ on P ( R ) satisfying µ ([0 , 1]) = 1 .

Theorem (G. Vitali 1905) There is no translation invariant measure µ on P ( R ) satisfying µ ([0 , 1]) = 1 . Theorem (F. Hausdorff 1914) There is no rotation invariant finitely additive probability measure on the sphere S 2 = { x ∈ R 3 : | x | = 1 } .

Theorem (G. Vitali 1905) There is no translation invariant measure µ on P ( R ) satisfying µ ([0 , 1]) = 1 . Theorem (F. Hausdorff 1914) There is no rotation invariant finitely additive probability measure on the sphere S 2 = { x ∈ R 3 : | x | = 1 } . Corollary There is no isometry invariant finitely additive measure µ on P ( R 3 ) such that µ ([0 , 1] 3 ) = 1 .

Theorem (G. Vitali 1905) There is no translation invariant measure µ on P ( R ) satisfying µ ([0 , 1]) = 1 . Theorem (F. Hausdorff 1914) There is no rotation invariant finitely additive probability measure on the sphere S 2 = { x ∈ R 3 : | x | = 1 } . Corollary There is no isometry invariant finitely additive measure µ on P ( R 3 ) such that µ ([0 , 1] 3 ) = 1 . Theorem (S. Banach 1923) There are isometry invariant finitely additive measures on P ( R ) and on P ( R 2 ) which are extensions of the Lebesgue measure.

A. Tarski, J. von Neumann, A. Haar and others: measures on groups, amenability, growth conditions etc.

A. Tarski, J. von Neumann, A. Haar and others: measures on groups, amenability, growth conditions etc. Theorem (S. Banach and A. Tarski 1924) If A , B ⊂ R k ( k ≥ 3) are bounded sets with nonempty interior, then A and B are equidecomposable.

A. Tarski, J. von Neumann, A. Haar and others: measures on groups, amenability, growth conditions etc. Theorem (S. Banach and A. Tarski 1924) If A , B ⊂ R k ( k ≥ 3) are bounded sets with nonempty interior, then A and B are equidecomposable. A and B are equidecomposable, if there are decompositions A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n s.t. A i is congruent to B i ( i = 1 , . . . , n ).

A. Tarski, J. von Neumann, A. Haar and others: measures on groups, amenability, growth conditions etc. Theorem (S. Banach and A. Tarski 1924) If A , B ⊂ R k ( k ≥ 3) are bounded sets with nonempty interior, then A and B are equidecomposable. A and B are equidecomposable, if there are decompositions A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n s.t. A i is congruent to B i ( i = 1 , . . . , n ). A. Tarski 1925 Is a disc in the plane equidecomposable with a square of the same area?

A. Tarski, J. von Neumann, A. Haar and others: measures on groups, amenability, growth conditions etc. Theorem (S. Banach and A. Tarski 1924) If A , B ⊂ R k ( k ≥ 3) are bounded sets with nonempty interior, then A and B are equidecomposable. A and B are equidecomposable, if there are decompositions A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n s.t. A i is congruent to B i ( i = 1 , . . . , n ). A. Tarski 1925 Is a disc in the plane equidecomposable with a square of the same area? Theorem (M.L. 1990) Yes.

Theorem (L. Grabowski, A. M´ ath´ e and O. Pikhurko 2015) If A , B ⊂ R k are bounded Borel sets with λ ( A ) = λ ( B ) > 0 and dim B ∂ A < k, dim B ∂ B < k, then A and B are equidecomposable using translations such that the pieces used in the decompositions are simultaneously Lebesgue measurable and have the Baire property.

Theorem (L. Grabowski, A. M´ ath´ e and O. Pikhurko 2015) If A , B ⊂ R k are bounded Borel sets with λ ( A ) = λ ( B ) > 0 and dim B ∂ A < k, dim B ∂ B < k, then A and B are equidecomposable using translations such that the pieces used in the decompositions are simultaneously Lebesgue measurable and have the Baire property. Theorem (A.S. Marks and S.T. Unger 2016) If A , B ⊂ R k are bounded Borel sets with λ ( A ) = λ ( B ) > 0 and dim B ∂ A < k, dim B ∂ B < k, then A and B are equidecomposable with finitely many Borel pieces and only using translations.

Theorem (I. Dubins, M. Hirsch, J. Karush 1963) The square and the disc are not “scissor-congruent”.

Theorem (I. Dubins, M. Hirsch, J. Karush 1963) The square and the disc are not “scissor-congruent”. Theorem (R.J. Gardner 1985) The square and the disc are not equidecomposable if the pieces are moved by a locally discrete group of isometries.

Theorem (I. Dubins, M. Hirsch, J. Karush 1963) The square and the disc are not “scissor-congruent”. Theorem (R.J. Gardner 1985) The square and the disc are not equidecomposable if the pieces are moved by a locally discrete group of isometries. A group G acting on R k is locally discrete, if every finitely generated subgroup H ⊂ G is discrete; i.e., if C ∩ g ( C ) = ∅ for all but a finitely many g ∈ H for every bounded C .

Theorem (I. Dubins, M. Hirsch, J. Karush 1963) The square and the disc are not “scissor-congruent”. Theorem (R.J. Gardner 1985) The square and the disc are not equidecomposable if the pieces are moved by a locally discrete group of isometries. A group G acting on R k is locally discrete, if every finitely generated subgroup H ⊂ G is discrete; i.e., if C ∩ g ( C ) = ∅ for all but a finitely many g ∈ H for every bounded C . R.J. Gardner 1988 Conjecture: if a polytope and a convex body are equidecomposable under isometries of an amenable group, then they are equidecomposable under the same isometries with convex pieces.

1−u 1 1−u R u 1 u E = { ( x , y ): ( x , y ) ∈ R } is a bipartite graph between [0 , 1] and [0 , 1]. M ⊂ V is a matching if it is the graph of a bijection of [0 , 1] onto itself.

1−u 1 1−u R u 1 u E = { ( x , y ): ( x , y ) ∈ R } is a bipartite graph between [0 , 1] and [0 , 1]. M ⊂ V is a matching if it is the graph of a bijection of [0 , 1] onto itself. Theorem (M.L. 1986) If u ∈ [0 , 1] \ Q then E contains a matching, but does not contain any matching which is Borel or measurable w.r.t. linear measure.

Let g 1 ( x ) = x + u , g 2 ( x ) = x − u , g 3 ( x ) = u − x , g 4 ( x ) = 1 − u − x . Then the g i are isometries of R , and they generate an amenable (in fact, solvable) group.

Let g 1 ( x ) = x + u , g 2 ( x ) = x − u , g 3 ( x ) = u − x , g 4 ( x ) = 1 − u − x . Then the g i are isometries of R , and they generate an amenable (in fact, solvable) group. R = [0 , 1] 2 ∩ � 4 i =1 graph g i .

Let g 1 ( x ) = x + u , g 2 ( x ) = x − u , g 3 ( x ) = u − x , g 4 ( x ) = 1 − u − x . Then the g i are isometries of R , and they generate an amenable (in fact, solvable) group. R = [0 , 1] 2 ∩ � 4 i =1 graph g i . Let M ⊂ R be a matching. If A i = π 1 ( M ∩ graph g i ), B i = π 2 ( M ∩ graph g i ), then [0 , 1] = A 1 ∪ . . . ∪ A 4 , [0 , 1] = B 1 ∪ . . . ∪ B 4 are decompositions and B i = g i ( A i ). So A = [0 , 1] and B = [0 , 1] are equidecomposable under isometries of an amenable group, but not equidecomposable under the same isometries with Borel pieces.

Let g 1 ( x ) = x + u , g 2 ( x ) = x − u , g 3 ( x ) = u − x , g 4 ( x ) = 1 − u − x . Then the g i are isometries of R , and they generate an amenable (in fact, solvable) group. R = [0 , 1] 2 ∩ � 4 i =1 graph g i . Let M ⊂ R be a matching. If A i = π 1 ( M ∩ graph g i ), B i = π 2 ( M ∩ graph g i ), then [0 , 1] = A 1 ∪ . . . ∪ A 4 , [0 , 1] = B 1 ∪ . . . ∪ B 4 are decompositions and B i = g i ( A i ). So A = [0 , 1] and B = [0 , 1] are equidecomposable under isometries of an amenable group, but not equidecomposable under the same isometries with Borel pieces. R.J. Gardner 1988 “If my conjecture is false, then the circle can be squared.”

A t ∼ B if there are decompositions A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n and vectors a 1 , . . . , a n s.t. B i = A i + a i ( i = 1 , . . . , n ).

A t ∼ B if there are decompositions A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n and vectors a 1 , . . . , a n s.t. B i = A i + a i ( i = 1 , . . . , n ). Theorem (M.L. 1992) If A , B ⊂ R k are bounded measurable sets with λ ( A ) = λ ( B ) > 0 and dim B ∂ A < k, dim B ∂ B < k, then A t ∼ B.

A t ∼ B if there are decompositions A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n and vectors a 1 , . . . , a n s.t. B i = A i + a i ( i = 1 , . . . , n ). Theorem (M.L. 1992) If A , B ⊂ R k are bounded measurable sets with λ ( A ) = λ ( B ) > 0 and dim B ∂ A < k, dim B ∂ B < k, then A t ∼ B. A t ∼ B ⇐ ⇒ ∃ φ : A → B bijection s.t. { φ ( x ) − x : x ∈ A } is finite.

A t ∼ B if there are decompositions A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n and vectors a 1 , . . . , a n s.t. B i = A i + a i ( i = 1 , . . . , n ). Theorem (M.L. 1992) If A , B ⊂ R k are bounded measurable sets with λ ( A ) = λ ( B ) > 0 and dim B ∂ A < k, dim B ∂ B < k, then A t ∼ B. A t ∼ B ⇐ ⇒ ∃ φ : A → B bijection s.t. { φ ( x ) − x : x ∈ A } is finite. If A = A 1 ∪ . . . ∪ A n , B = B 1 ∪ . . . ∪ B n and B i = A i + a i ( i = 1 , . . . , n ),