An example against the periodic tiling assumption

An example against the periodic tiling assumption

Rachel Greenfeld and I simply uploaded our article on arXiv”An example against the periodic tiling assumption“. That is the total model of the consequence I introduced on this weblog just a few months in the past. periodic tiling assumption associated to Grunbaum-Shephard And Lagarias-Wang. The article took a bit longer than anticipated as a consequence of a technical problem that we weren’t conscious of on the time of the announcement and wanted a workaround.

In additional element: The unique technique, as described within the announcement, was to create a “tiling language” that might encode a specific “sequence”.-adic Sudoku puzzle ”after which present that the second kind of puzzle has non-periodic options provided that: {P} was a big sufficient prime quantity. It appears that evidently the second half of this technique labored, however there was an issue with the primary half: Our tile language (utilizing this) {2}-group valued capabilities) to encode arbitrary boolean relationships between boolean capabilities and in addition (utilizing) {{bf Z}/p{bf Z}}-value capabilities) to encode “clock” capabilities akin to {n mapsto n hbox{ mod } p} it was part of us {P}-adic Sudoku puzzle, however we have not been in a position to get these two sorts of capabilities to “discuss” to one another as wanted to code. {P}-adic Sudoku puzzle (primary downside, if {H} a finite commutative {2}-group then there aren’t any non-trivial subgroups. {H times {bf Z}/p{bf Z}} not included in {H} or insignificant {{bf Z}/p{bf Z}} course). Consequently, we needed to change our “”.{P}-adic Sudoku puzzle” by “{2}-adic Sudoku puzzle” mainly means changing the prime {P} by a sufficiently giant power {2} (we imagine {2^{10}} will suffice). This fastened the encoding problem, however {2}-adic Sudoku puzzles had been a bit extra complicated than others. {P}-adic situation, because of the following motive. Beneath is a pleasant evaluation train:

Theorem 1 (Linearity in three instructions means full linearity) To offer permission {F: {bf R}^2 rightarrow {bf R}} be an everyday perform that’s affine-linear on each horizontal line, diagonal (slope line) {one}) and anti-diagonal (slope line {-one}). In different phrases, for any {c in {bf R}}capabilities {x mapsto F(x,c)}, {x mapsto F(x,c+x)}And {x mapsto F(x,cx)} each affine perform {{bf R}}. Later {F} is an affine perform on {{bf R}^2}.

Certainly, the property of being affine in three instructions signifies that there’s the second-order type related to Hessian. {nabla^2 F(x,y)} disappears at any level {(1,0)}, {(1,1)}And {(1,-1)}and so it ought to disappear all over the place. In actual fact, the smoothness speculation isn’t essential; We go away this as an train to the reader. If one is substituted, the identical expression seems to be true. {{bf R}} with cyclic group {{bf Z}/p{bf Z}} if {P} unusual; that is the important thing for us to indicate {P}-adic Sudoku puzzles have a (roughly) two-dimensional affinity construction, which can be utilized later in additional evaluation to indicate that they’re truly non-periodic. Nonetheless, the corresponding declare for cyclic teams {{bf Z}/q{bf Z}} when can it fail {Q} is powerful sufficient {2}! Really the overall type of capabilities {F: ({bf Z}/q{bf Z})^2 rightarrow {bf Z}/q{bf Z}} takes the affine type on every horizontal line, diagonal, and diagonal

displaystyle F(x,y) = Ax + By + C + D frac{q}{4} y(xy)

for some integer coefficients {A B C D}. This extra time period “pseudo-affine” {D frac{q}{4} y(xy)} causes some extra technical issues, however finally seems to be manageable.

Throughout the writing course of, we additionally found that the coding a part of the proof turns into extra modular and conceptual when two new definitions, “expressible property” and “poorly expressible property”, are added. These ideas are considerably just like: {Pi^0_0} sentences and {Sigma^0_1} sentences in arithmetic hierarchyor algebraic sets And semi-algebraic sets in actual algebraic geometry. Roughly talking, an expressible property is a property of a set of capabilities. {f_w: G rightarrow H_w}, {w in {mathcal W}} from a commutative group {G} finite abelian teams {h_w}such that the characteristic may be expressed on the graph by way of a number of tiling equations

displaystyle A := { (x, (f_w(x))_{w in {mathcal W}} subset G times prod_{w in {mathcal W}} H_w.

For instance, the property that two capabilities have {f,g: {bf Z} rightarrow H} the distinction with respect to a continuing may be expressed by way of the tiling equation

displaystyle A oplus ({0} times H^2) = {bf Z} times H^2

(vertical line check) and in addition

displaystyle A oplus ({0} times Delta cup {1} times (H^2 backslash Delta)) = G times H^2,

The place {Delta = {(h,h): h in H }} is the diagonal subgroup {H^2}. A poorly expressible characteristic {P} is an existential measurement of some expressible trait {P^*}so a set of capabilities {(f_w)_{w in W}} suits property {P} if and provided that there’s an extension of this tuple by some extra performance that matches the characteristic {P^*}. It seems that weak expressible options are turned off below a collection of helpful operations, permitting us to simply create fairly complicated weakly expressible options from a “library” of easy weakly expressible options, akin to a fancy pc program may be created. easy library routines. Specifically, we can “program” our Sudoku puzzle as a weakly expressive characteristic.

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