Pure Mathematics

Motivation

The third mission in the series formalizing Allen Hatcher's Algebraic Topology (Cambridge University Press, 2002; pi.math.cornell.edu/~hatcher/AT/AT.pdf) turns to the second main topic of Chapter 1, covering spaces (Section 1.3, pp. 56–78). The first mission used the covering $\mathbb{R}\to S^1$ to compute $\pi_1(S^1)$, and the second proved van Kampen's theorem. This mission develops the general theory of covering spaces of a fixed space $X$: the lifting properties (pp. 60–62), the classification of connected covering spaces by subgroups of $\pi_1(X)$ (pp. 63–68), and deck transformations and group actions (pp. 70–72). Its goal is the classification theorem (Theorem 1.38, p. 67), Hatcher's "Galois correspondence" between path-connected covering spaces of $X$ and subgroups of $\pi_1(X,x_0)$, together with its companions Proposition 1.39 (deck groups and normal covers) and Proposition 1.40 (covering space actions and orbit spaces).

All statements live in the Lean namespace Hatcher used by the earlier missions.

Setting

A covering space of $X$ (p. 56) is a space $\tilde X$ with a map $p:\tilde X\to X$ such that every $x\in X$ has an open neighborhood $U$ whose preimage is a disjoint union of open sets each mapped homeomorphically onto $U$; $p^{-1}(U)$ may be empty, so $p$ need not be surjective. This is Mathlib's IsCoveringMap. For a covering space with basepoints $p:(\tilde X,\tilde x_0)\to(X,x_0)$ we write

for the induced homomorphism (Hatcher.coverHom) and its image (Hatcher.coverSubgroup).

$X$ is semilocally simply-connected (p. 63, Hatcher.IsSemilocallySimplyConnected) if each $x\in X$ has a neighborhood $U$ such that every loop at $x$ contained in $U$ is null-homotopic in $X$. The bundle Hatcher_Covering also fixes: the structure CoveringSpace X (a total space $\tilde X$ and a covering map $p$) and its pointed version PointedCover X x₀ (with $\tilde x_0\in p^{-1}(x_0)$ and associated subgroup PointedCover.subgroup); isomorphism of covering spaces (p. 67), a homeomorphism $f:\tilde X_1\to\tilde X_2$ with $p_1=p_2f$, with or without preservation of basepoints (IsIsomorphic, IsPointedIsomorphic); the deck transformation group $G(\tilde X)$ (p. 70, deckGroup), the self-homeomorphisms of $\tilde X$ commuting with $p$; normal covering spaces (p. 70, IsNormalCover); Hatcher's condition $(\ast)$ for a covering space action of a group $G$ on $Y$ (p. 72, IsCoveringSpaceAction); and the orbit space $Y/G$ with its quotient map (OrbitSpace, orbitProj).

Formalization targets

Goal (Theorem 1.38, p. 67)

Let $X$ be path-connected, locally path-connected and semilocally simply-connected, with basepoint $x_0$. Then:

1. every subgroup $H\le\pi_1(X,x_0)$ is $p_*\pi_1(\tilde X,\tilde x_0)$ for some path-connected covering space with basepoint;
2. two path-connected covering spaces with basepoints are isomorphic by a basepoint-preserving isomorphism iff their subgroups coincide;
3. two path-connected covering spaces are isomorphic (basepoints ignored) iff their subgroups, at some choice of basepoints over $x_0$, are conjugate in $\pi_1(X,x_0)$.

Together these say that $(\tilde X,\tilde x_0)\mapsto p_*\pi_1(\tilde X,\tilde x_0)$ is a bijection from basepoint-preserving isomorphism classes of path-connected covering spaces to subgroups, inducing a bijection from isomorphism classes to conjugacy classes of subgroups.

Milestones

1. Proposition 1.31 (p. 61), first part: $p_*$ is injective.
2. Proposition 1.31, second part: $p_*\pi_1(\tilde X,\tilde x_0)$ consists of the classes of loops at $x_0$ whose lifts starting at $\tilde x_0$ are loops.
3. Proposition 1.32 (p. 61): for $X,\tilde X$ path-connected, the fibre $p^{-1}(x_0)$ is in bijection with the cosets of $H$, so the number of sheets is the index of $H$.
4. Proposition 1.33 (p. 61), the lifting criterion: for $Y$ path-connected and locally path-connected, $f:(Y,y_0)\to(X,x_0)$ lifts to $(\tilde X,\tilde x_0)$ iff $f_*\pi_1(Y,y_0)\subseteq H$.
5. Proposition 1.34 (p. 62), unique lifting: two lifts of $f:Y\to X$ agreeing at one point agree everywhere if $Y$ is connected.
6. Necessity of semilocal simple connectivity (p. 63): if $X$ has a simply-connected covering space (surjective onto $X$), then $X$ is semilocally simply-connected.
7. Existence of a simply-connected covering space (pp. 63–65): if $X$ is path-connected, locally path-connected and semilocally simply-connected, it has a simply-connected covering space (the universal cover).
8. Proposition 1.36 (p. 66): under the same hypotheses, every subgroup $H\le\pi_1(X,x_0)$ is realized as $p_*\pi_1(X_H,\tilde x_0)$ for a path-connected covering space.
9. Proposition 1.37 (p. 67): for $X$ path-connected and locally path-connected, two path-connected covering spaces with basepoints are basepoint-preservingly isomorphic iff their subgroups are equal.
10. Change of basepoint (pp. 67–68, proof of Theorem 1.38): moving $\tilde x_0$ within $p^{-1}(x_0)$ replaces $H$ by a conjugate, and every conjugate arises this way.
11. Proposition 1.39(a) (p. 71): a path-connected covering space of a path-connected, locally path-connected $X$ is normal iff $H$ is a normal subgroup.
12. Proposition 1.39(b): $G(\tilde X)\cong N(H)/H$, given as a surjective homomorphism $N(H)\to G(\tilde X)$ with kernel $H$.
13. Proposition 1.39, final clause: for the universal cover, $G(\tilde X)\cong\pi_1(X,x_0)$.
14. Proposition 1.40(a) (p. 72): for a covering space action of $G$ on $Y$, the quotient map $Y\to Y/G$ is a normal covering space.
15. Proposition 1.40(b): if moreover $Y$ is path-connected, $G$ is the group of deck transformations of $Y\to Y/G$, via $g\mapsto(y\mapsto gy)$.
16. Proposition 1.40(c): if $Y$ is path-connected and locally path-connected, $G\cong\pi_1(Y/G)/p_*\pi_1(Y)$, given as a surjective homomorphism $\pi_1(Y/G)\to G$ with kernel $p_*\pi_1(Y)$.

Significance

The result itself. The classification theorem is the central structural fact about covering spaces: the connected coverings of $X$ are "the same as" the subgroups of $\pi_1(X)$, with the universal cover corresponding to the trivial subgroup and normal coverings to normal subgroups. Proposition 1.40 is the standard method for computing fundamental groups of orbit spaces ($\pi_1(\mathbb{RP}^n)=\mathbb{Z}/2$, $\pi_1(T^n)=\mathbb{Z}^n$, lens spaces) and is used throughout Hatcher's later chapters.

Formalizing it. Mathlib (at this environment's revision) already has the lifting theory for covering maps: path and homotopy lifting (IsCoveringMap.liftPath, liftHomotopy), the monodromy action (IsCoveringMap.monodromy), the injectivity of $p_*$ (injective_path_homotopic_map, cited there as Proposition 1.31), the unique-lifting statement (IsCoveringMap.eq_of_comp_eq), and the lifting criterion itself (existsUnique_continuousMap_lifts_of_range_le, cited as Proposition 1.33). For quotient maps by a free properly discontinuous action it has IsQuotientCoveringMap, with the homomorphism $\pi_1(Y/G)\to G^{\mathrm{op}}$ and its kernel and surjectivity. Milestones 1, 4, 5 and 16 are therefore expected to be short reductions to Mathlib, and Mathlib's quotient-covering theory should carry most of milestones 14–15. Mathlib has no notion of semilocal simple connectivity, no construction of the universal cover or of the coverings $X_H$, no classification theorem, and no deck transformation groups or normal coverings; milestones 6–13 and the goal are new.

Difficulty

The heart of the mission is the construction of the universal cover (pp. 63–65): the points are homotopy classes of paths from $x_0$, the topology is generated by the sets $U_{[\gamma]}$ for $U$ in the basis of path-connected open sets on which $\pi_1$ dies, and one must verify that this is a topology basis, that $p$ is a covering map, and that the result is simply connected. Proposition 1.36 then passes to a quotient by $H$ and checks that the projection remains a covering map. Both are elementary but long, and formalizing them requires a systematic treatment of path homotopy classes as points of a space.

Propositions 1.37 and 1.39 follow from the lifting criterion and unique lifting; the deck-group homomorphism in 1.39(b) sends a loop in $N(H)$ to the deck transformation produced by the lifting criterion, and its kernel is computed by Proposition 1.31. Proposition 1.40(a) needs the quotient topology on $Y/G$ and the evenly covered neighborhoods $p(U)$ from condition $(\ast)$; part (b) is the observation that a deck transformation of a path-connected cover is determined by one value. Milestone 3 is orbit–stabilizer for the monodromy action.

Formalization scope

• Spaces are arbitrary topological spaces; hypotheses (path-connectedness, local path-connectedness, semilocal simple connectivity, connectedness of the domain in Proposition 1.34) are stated per theorem, exactly where Hatcher assumes them.
• CoveringSpace X bundles a total space in the same universe as $X$ with a covering map; the classification quantifies over covering spaces in this sense. Since the universal cover and the coverings $X_H$ are constructed from paths in $X$, they live in that universe, so nothing is lost.
• "Isomorphic" is the existence of a homeomorphism over $X$ (Hatcher, p. 67), a proposition on pairs of covering spaces; Theorem 1.38 is stated as the three-part conjunction above rather than as a bijection between quotient sets, which avoids forming the set of isomorphism classes of types while asserting exactly the same content.
• Conjugacy is expressed with Mathlib's MulAut.conj; "number of sheets equals the index" is stated as a bijection $p^{-1}(x_0)\simeq\pi_1(X,x_0)/H$ with the coset space.
• The isomorphisms of Propositions 1.39(b) and 1.40(c) are stated as surjective homomorphisms with prescribed kernel, which is how Hatcher proves them and avoids requiring a Normal instance in the statement; the final clause of 1.39 and 1.40(b) are stated as the existence of group isomorphisms, the latter with its action on $Y$ prescribed.
• A covering space action includes continuity of each $y\mapsto gy$ (Hatcher's actions are by homeomorphisms). The orbit space is Mathlib's MulAction.orbitRel.Quotient with the quotient topology.
• Trivializing readings are excluded: path-connected spaces are nonempty, and a nonempty covering space of a path-connected base is surjective; milestone 6 assumes surjectivity explicitly because its base need not be connected.

Contributions welcome: a reusable construction of the space of path classes with its topology, the covering $X_H$, and the deck-group homomorphism; the short reductions to Mathlib for Propositions 1.31, 1.33 and 1.34 are good first contributions.

Selected references

• A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Section 1.3, pp. 56–72. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
• E. H. Spanier, Algebraic Topology, Springer, 1966, Chapter 2 (covering spaces and the classification theorem).
• J. R. Munkres, Topology, 2nd ed., Prentice Hall, 2000, Chapter 13 (classification of covering spaces).
• Mathlib, Mathlib/Topology/Covering/Basic.lean (covering maps). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/Topology/Covering/Basic.lean
• Mathlib, Mathlib/Topology/Homotopy/Lifting.lean (path and homotopy lifting, monodromy, the lifting criterion). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/Topology/Homotopy/Lifting.lean
• Mathlib, Mathlib/Topology/Covering/Quotient.lean (quotient covering maps for group actions). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/Topology/Covering/Quotient.lean

Motivation

Once $\pi_1(S^1)\cong\mathbb{Z}$ is known, the next question in Allen Hatcher's Algebraic Topology (Cambridge University Press, 2002; pi.math.cornell.edu/~hatcher/AT/AT.pdf) is how to compute fundamental groups of spaces built from pieces. Section 1.2 answers this with van Kampen's theorem (Theorem 1.20, p. 43): if a space is covered by open sets with a common basepoint and path-connected intersections, its fundamental group is the free product of the fundamental groups of the pieces, modulo relations coming from the intersections. It is the main computational tool of Chapter 1: it gives the fundamental groups of wedges of circles, graphs, surfaces, and every CW complex from its 2-skeleton (Propositions 1.26–1.28), and it underlies the classification of covering spaces in Section 1.3.

This is the second mission in the series formalizing Hatcher's book. The first mission established the covering-space lifting properties and $\pi_1(S^1,1)\cong\mathbb{Z}$ in the Lean namespace Hatcher; this one covers the subsection "The van Kampen Theorem" of Section 1.2 (pp. 41–47) together with its warm-up Lemma 1.15 and Proposition 1.14 from Section 1.1 (p. 35).

Setting

Let $X$ be a topological space with a basepoint $x_0$. A path is a continuous map $I=[0,1]\to X$, a loop at $x_0$ is a path with both endpoints $x_0$, and $\pi_1(X,x_0)$ is the group of homotopy classes of loops at $x_0$ under concatenation. A continuous map $\varphi:X\to Y$ with $\varphi(x_0)=y_0$ induces a homomorphism $\varphi_*:\pi_1(X,x_0)\to\pi_1(Y,y_0)$, $[f]\mapsto[\varphi\circ f]$.

Let $(A_\alpha)_{\alpha\in\iota}$ be a family of subsets of $X$, each containing $x_0$, with the subspace topology; write $\pi_1(A_\alpha)$ for $\pi_1(A_\alpha,x_0)$. The inclusions $A_\alpha\hookrightarrow X$ induce

which are Hatcher.inclHom, and the inclusions $A_\alpha\cap A_\beta\hookrightarrow A_\alpha$ and $A_\alpha\cap A_\beta\hookrightarrow A_\beta$ induce

which are Hatcher.interHomLeft and Hatcher.interHomRight.

The free product $\ast_\alpha G_\alpha$ of a family of groups is the group of reduced words in the $G_\alpha$ (Hatcher, pp. 41–42); in Lean it is Mathlib's Monoid.CoprodI, here Hatcher.FreeProd. Its universal property extends the $j_\alpha$ to a single homomorphism

Hatcher.vanKampenHom. Since $j_\alpha i_{\alpha\beta}=j_\beta i_{\beta\alpha}$ (both are induced by $A_\alpha\cap A_\beta\hookrightarrow X$), the elements

lie in the kernel of $\Phi$. Let $N$ be the normal subgroup generated by all of them, Hatcher.vanKampenNormal.

Formalization targets

Goal (Theorem 1.20)

If $X$ is the union of path-connected open sets $A_\alpha$ each containing $x_0$, each $A_\alpha\cap A_\beta$ is path-connected, and each $A_\alpha\cap A_\beta\cap A_\gamma$ is path-connected, then

Hence $\Phi$ induces an isomorphism $\pi_1(X)\cong\ast_\alpha\pi_1(A_\alpha)/N$.

Milestones

1. Lemma 1.15 (p. 35). If $X$ is the union of path-connected open sets $A_\alpha$ containing $x_0$ with each $A_\alpha\cap A_\beta$ path-connected, then every loop in $X$ at $x_0$ is homotopic to a product of loops each of which is contained in a single $A_\alpha$.
2. Proposition 1.14 (p. 35). $\pi_1(S^n)=0$ for $n\ge 2$.
3. Theorem 1.20, first part (p. 43). Under the hypotheses of Lemma 1.15, $\Phi$ is surjective.
4. The kernel contains the relators (p. 43). $N\le\ker\Phi$, with no hypotheses on the cover.
5. Theorem 1.20, second part (p. 43). If moreover every triple intersection is path-connected, $\ker\Phi\le N$.
6. Induced isomorphism (p. 43). Under the same hypotheses there is an isomorphism $\ast_\alpha\pi_1(A_\alpha)/N\cong\pi_1(X)$ sending the class of a word to its image under $\Phi$.

Significance

The result itself. Van Kampen's theorem is the gluing law for $\pi_1$. With it Hatcher computes $\pi_1$ of wedge sums (free products), of graphs (free groups), of the closed orientable surfaces (Example 1.26), and shows that attaching $2$-cells kills exactly the attaching loops (Proposition 1.26), so that every group is a fundamental group (Corollary 1.28). The surjectivity half alone gives Proposition 1.14, that spheres of dimension at least two are simply connected, and hence that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^n$ for $n\ne 2$ (Corollary 1.16).

Formalizing it. Mathlib has the fundamental groupoid and fundamental group, induced homomorphisms (FundamentalGroup.map), free products of groups (Monoid.CoprodI) with their universal property, normal closures, and quotient groups. It has no version of van Kampen's theorem for topological spaces (its CategoryTheory/Limits/VanKampen concerns colimits in categories, not fundamental groups), and no computation of $\pi_1(S^n)$ for $n\ge 2$; on the platform, however, the theorem SP4Mission.sphere_simplyConnected (already proved in this environment) states that the unit sphere of $\mathbb{R}^n$ is simply connected for $n\ge 3$, which is Proposition 1.14 with shifted indexing, so that milestone can be closed by a one-line reduction. The mission supplies the statements in Hatcher's form, on Mathlib's $\pi_1$, so that later missions (covering spaces, cell complexes) can use them directly.

Difficulty

Surjectivity is a compactness argument: subdivide $I$ so each piece of the loop lies in one $A_\alpha$, then use path-connectedness of the intersections to connect the subdivision points back to $x_0$. The formal difficulty is bookkeeping: producing the subdivision from an open cover of $[0,1]$ (Mathlib's exists_monotone_Icc_subset_open_cover_unitInterval is the tool) and showing the reparametrised concatenation is homotopic to the original loop.

The kernel computation is the hard part. Hatcher's proof takes a homotopy $F:I\times I\to X$ between two factorizations, subdivides the square into rectangles each mapped into a single $A_\alpha$, perturbs the grid so at most three rectangles meet at a corner (this is where triple intersections enter), and then shows that moving the loop across one rectangle at a time changes the factorization only by the two elementary moves that hold in $\ast_\alpha\pi_1(A_\alpha)/N$. Every step is elementary, but the induction over the grid is long, and each elementary move requires an explicit path-homotopy in a subspace. The naive idea of proving $\ker\Phi\le N$ by an induction on word length does not work: the relation between two factorizations of the same loop is only visible through a homotopy in $X$, not through the words.

Proposition 1.14 is easy given Lemma 1.15 but requires exhibiting the cover of $S^n$ by two complements of antipodal points, showing each is simply connected (homeomorphic to $\mathbb{R}^n$ via stereographic projection, which Mathlib has as stereographic), and showing their intersection is path-connected when $n\ge 2$.

Formalization scope

• The index set $\iota$ and the space $X$ are arbitrary; the $A_\alpha$ are Set X with the subspace topology, and $\pi_1(A_\alpha)$ is Mathlib's FundamentalGroup ↥(A α) ⟨x₀, _⟩. Hypotheses are stated explicitly on each theorem: IsOpen, IsPathConnected, ⋃ α, A α = Set.univ, and path-connectedness of pairwise (and, where Hatcher requires it, triple) intersections.
• $i_{\alpha\beta}$ and $i_{\beta\alpha}$ are both defined on $\pi_1(A_\alpha\cap A_\beta)$ (rather than on $\pi_1(A_\beta\cap A_\alpha)$ for the second), so no identification of $A_\alpha\cap A_\beta$ with $A_\beta\cap A_\alpha$ is needed; the set of relators ranges over all ordered pairs $(\alpha,\beta)$.
• "Product of loops" in Lemma 1.15 is a finite List of loops, each tagged with the index $\alpha$ of the piece it lies in, concatenated right-to-left with the constant loop as empty product (Hatcher.loopProd). Any bracketing gives the same homotopy class.
• The goal is stated as the conjunction "surjective and $\ker\Phi=N$"; the isomorphism $\ast_\alpha\pi_1(A_\alpha)/N\cong\pi_1(X)$ is a separate milestone, stated as the existence of a group isomorphism compatible with $\Phi$ on the quotient, which pins it down uniquely.
• $S^n$ is Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+1))) 1, and "$\pi_1(S^n)=0$" is Mathlib's SimplyConnectedSpace (path-connected with trivial fundamental group), which is what Hatcher means since $S^n$ is path-connected.
• Trivializing readings are excluded: the cover hypotheses do not force $\iota$ nonempty, but then $X=\bigcup A_\alpha=\varnothing$ contradicts the existence of $x_0$, so the statements are not vacuous in any interesting case, and $\Phi$ is the specific homomorphism induced by the inclusions.

Contributions welcome: the subdivision lemma for loops in an open cover, a reusable treatment of factorizations and their elementary moves, and the two-set special case $\pi_1(X)\cong(\pi_1(A)\ast\pi_1(B))/N$ as a corollary.

Selected references

• A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Section 1.2, pp. 40–49; Lemma 1.15 and Proposition 1.14, p. 35. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
• E. R. van Kampen, On the connection between the fundamental groups of some related spaces, American Journal of Mathematics 55 (1933), 261–267. https://doi.org/10.2307/2371128
• H. Seifert, Konstruktion dreidimensionaler geschlossener Räume, Berichte Sächs. Akad. Leipzig 83 (1931), 26–66.
• Mathlib, Mathlib/GroupTheory/CoprodI.lean (free products of groups). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/GroupTheory/CoprodI.lean
• Mathlib, Mathlib/AlgebraicTopology/FundamentalGroupoid/FundamentalGroup.lean (fundamental group and induced homomorphisms). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/AlgebraicTopology/FundamentalGroupoid/FundamentalGroup.lean

Motivation

The Hardy--Littlewood circle method is the principal analytic tool for counting solutions to additive equations in integers. Introduced by Hardy and Ramanujan for the partition function and developed by Hardy and Littlewood in their Partitio Numerorum series (1920--1928), it produces asymptotic formulae for the number of representations of a large integer $n$ as a sum of $s$ terms drawn from a prescribed set — $k$-th powers, primes, values of a polynomial.

Its engine is an estimate for exponential sums. If a sum $\sum_{x<N} e(\alpha x^k)$, where $e(\theta)=\exp(2\pi i\theta)$, exhibits cancellation for every $\alpha$ not well approximable by a rational with small denominator, the method delivers an asymptotic formula; if it does not, the method stalls. Weyl's inequality (Weyl 1916) was the first such estimate and remains the standard one for moderate $k$.

A short timeline of the estimate this mission targets. Weyl (1916) proved the inequality below with the exponent $2^{1-k}$, in the course of his work on uniform distribution. Hardy and Littlewood (1920--1928) built the circle method on it, obtaining $G(k)\le (k-2)2^{k-1}+5$ for Waring's problem. Vinogradov (1935) replaced Weyl differencing by his mean value theorem, superior for large $k$, reducing the bound to $O(k\log k)$; Wooley's efficient congruencing (2012) and the Bourgain--Demeter--Guth decoupling theorem (2016) settled the main conjecture of Vinogradov's mean value theorem. For small $k$ — and as the entry point to the subject — Weyl's inequality is still the right tool, and it is the natural first capstone for a formalization of the method.

Setting

For a real number $\theta$ write

the standard additive character of $\mathbb{R}/\mathbb{Z}$: it satisfies $e(x+y)=e(x)e(y)$, $|e(x)|=1$, and $e(x)=1$ exactly when $x\in\mathbb{Z}$.

For a real number $\theta$ write $\|\theta\|$ for the distance from $\theta$ to the nearest integer. It is periodic with period $1$, vanishes exactly on $\mathbb{Z}$, satisfies the triangle inequality, and is at most $\tfrac12$.

Given a finite set $A\subseteq\mathbb{Z}$, its generating function is $f_A(\theta)=\sum_{a\in A}e(a\theta)$. The basic identity of the subject is

a consequence of the orthogonality relation $\int_0^1 e(m\theta)\,d\theta=[\,m=0\,]$.

A Weyl sum of degree $k$ is $\sum_{0\le x<N} e(\alpha x^k)$. The whole difficulty is to bound it for $\alpha$ in the minor arcs — those $\alpha$ admitting no rational approximation $a/q$ with $q$ small.

Target

Fix $k\ge 2$. For every $\varepsilon>0$ there is a constant $C=C(k,\varepsilon)$ such that whenever $(a,q)=1$, $q\ge 1$, and $\left|\alpha-\frac{a}{q}\right|\le \frac{1}{q^2}$,

The intermediate targets, weakest first, are the milestone list: the counting identity, the Weyl differencing (squaring) step, the Farey covering with coprime numerator, the divisor bound $d(n)\ll_\varepsilon n^\varepsilon$, Hua's fourth-moment inequality for $k=2$, and the degree-two case of the inequality itself.

Significance

The result itself. Weyl's inequality is what makes the minor arcs negligible. Applied with $q$ in the range $N^{\delta}\le q\le N^{k-\delta}$ it gives a power saving over the trivial bound $N$, and integrating that saving over the minor arcs shows their contribution is smaller than the main term produced by the major arcs. Every classical application of the circle method — the asymptotic formula in Waring's problem, Vinogradov's three primes theorem, the Birch--Davenport theory of forms in many variables — passes through an estimate of this shape. Without it the method produces an identity, not a theorem.

Formalizing it. Mathlib currently contains the analytic prerequisites — Fourier characters on AddCircle, Dirichlet's approximation theorem, Abel summation, Gauss sums — but no circle-method apparatus whatsoever: no Weyl sums, no arc dissection, no singular series, no mean value estimates. This mission supplies the first layer. The foundational tier is already machine-checked: 53 theorems covering the character $e$, the norm $\|\cdot\|$, the geometric sum bound $\left|\sum_{x<N}e(x\theta)\right|\le\min\!\left(N,\frac{1}{2\|\theta\|}\right)$, both orthogonality relations, both forms of Dirichlet's theorem, and the basic theory of $f_A$, are published on the platform with verified proofs and may be imported freely. What remains open is the combinatorial and analytic core listed in the milestones. None of the milestone statements is currently formalized anywhere, to the best of our knowledge.

Difficulty

The obvious approach fails immediately. One would like to sum $\left|\sum_{x<N}e(\alpha x^k)\right|$ by comparing it to the linear case, where the geometric series gives $\min(N,\frac{1}{2\|\alpha\|})$ outright. But for $k\ge2$ the summand is not a geometric progression and there is no closed form.

Weyl's device is to square and difference: $\left|\sum_x e(\phi(x))\right|^2=\sum_{x,y}e(\phi(x)-\phi(y))$, and the substitution $y=x+h$ turns the inner polynomial into one of degree $k-1$ in $x$. Iterating $k-1$ times reduces to a linear sum, at the cost of raising the estimate to the power $2^{1-k}$ — which is why the saving is so weak for large $k$, and why Vinogradov's method eventually supersedes it.

The genuine obstacles in a formalization are: (i) bookkeeping the shifted ranges produced by each differencing step, which are not $[0,N)$ and must be handled uniformly; (ii) the divisor bound $d(n)\ll_\varepsilon n^\varepsilon$, needed to count the $h$ for which the resulting linear coefficient is close to an integer, and which is not currently in Mathlib in this form; (iii) tracking the $\varepsilon$-dependent constants through $k-1$ iterations without the informal $\ll$ notation.

Formalization scope

Statements are given over the Prove2Me default environment (Lean v4.30.0, Mathlib c5ea003), in the shared namespace CircleMethod, and build on two published definitions: CircleMethod_char (the character e and the norm nrm) and CircleMethod_genfun (the generating function f).

Conventions this mission commits to:

• $\|\theta\|$ is nrm θ = |θ - round θ|. Mathlib's round breaks ties upwards, so round is not an odd function; the characterisation to use is minimality, nrm θ ≤ |θ - n| for every integer n, which is published as CircleMethod.nrm_le.
• Sums run over Finset.range N, that is $0\le x<N$, and $N$ is a natural number. Hypotheses 0 < N and 0 < q are stated explicitly rather than left implicit.
• Asymptotic notation is eliminated in favour of explicit existential constants: $X\ll_\varepsilon Y$ is rendered as ∀ ε > 0, ∃ C > 0, ∀ …, X ≤ C * Y, with the constant quantified outside the parameters it may depend on and inside nothing else. Solvers should not weaken this by allowing $C$ to depend on $N$, $q$ or $\alpha$.
• Exponents such as $N^{1+\varepsilon}$ and $2^{1-k}$ are real powers (Real.rpow), not natural powers.
• Coprimality is Nat.Coprime a.natAbs q, which is the correct notion for a possibly negative numerator.

One trivialising formalization to rule out: the goal must not be read with $C$ permitted to depend on $N$, since then $C=N$ makes it vacuous. The quantifier order in the Lean statement already forbids this, and solvers should preserve it exactly.

Contributions welcome on any milestone independently; the divisor bound and the Farey covering are self-contained and need no other milestone. Both are reusable well beyond this mission.

Selected references

• H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Mathematische Annalen 77 (1916), 313--352. DOI:10.1007/BF01475864
• G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio Numerorum' I--VI, 1920--1928.
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