{"id":"9701ed3e-1ee5-4fa7-8af3-c6980baead0f","arxiv_id":"2412.05858","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For all m,n with m+n>2 and any norms, the set of possible asymptotic Dirichlet approximation errors is exactly [0,Delta], with every value attained densely and uncountably often.","lead":"This paper proves that for every dimension pair (m,n) except (1,1), the Dirichlet approximation spectrum of m-by-n matrices is the full interval from 0 to the Dirichlet constant, for any norms. The result settles a question studied in a string of recent papers and gives a general topological criterion for when such spectra are intervals.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5's proof relies on 'nested open sets with compact closures have nonempty intersection,' which is false; without enforcing cl(Ω_{k+1})⊂Ω_k, the constructed point x0 may not exist, so Theorems 2 and 1 are not yet proved.","rationale":"I read the paper in good faith. The central claim, Theorem 1, is derived from Theorem 2, which in turn is an application of Theorem 5. The proof of Theorem 5 contains a concrete and clearly false assertion: nested nonempty open sets with compact closures need not have nonempty intersection (e.g., Ω_k = (0,1/k) in R). This invalidates the construction of the point x0 and therefore the proof of Theorems 2 and 1 as written. I agree with the reader that this is a real gap requiring repair. I do not share the reader's emphasis on the Kleinbock-Weiss equidistribution theorem as an equally load-bearing concern: for Theorem 1 only the uniform weights (1/m,...,1/m; 1/n,...,1/n) are needed, and the quoted equidistribution result is a published theorem whose hypotheses appear satisfied; even for Theorem 2, the statement as quoted is plausible given the expanding horospherical structure. The closure-nesting gap, by contrast, is internal and affects every application. The fix is straightforward — choose U_w with compact closure inside Ω_k and then take Ω_{k+1} inside U_w, so that cl(Ω_{k+1}) ⊂ Ω_k and Cantor's theorem applies. Because this repair is local and does not alter the overall strategy, I retain the CONDITIONAL verdict rather than moving to reject or accept. The paper's strategy is coherent, the reduction to the topological theorem is well organized, and the applications of equidistribution are standard; the main obstacle is the incomplete induction in Theorem 5.","tokens_in":21179,"tokens_out":24559,"duration_ms":248319,"concrete_test":"Re-derive the induction in Theorem 5 with the additional invariant cl(Ω_{k+1}) ⊂ Ω_k. Concretely, at the step choosing U_w, use local compactness (which the paper derives from local connectedness via (X_n)) to pick U_w with compact closure inside Ω_k while preserving the inequalities (12); then check that the local-connectedness property supplies Ω_{k+1} with compact closure inside U_w ∩ Ω_k, and that conditions (C),(D),(E) are unaffected. If this verification succeeds, the proof is patchable and the reader's CONDITIONAL verdict stands. If it fails, exhibit a counterexample satisfying the hypotheses of Theorem 5 with empty ∩ Ω_k.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction is Theorem 5. In its proof, after building a sequence (Ω_k) of nonempty open sets satisfying Ω_k ⊂ Ω_{k−1} and each having compact closure, the text states: 'Ω_k is non-empty and open with compact closure for all k, hence the intersection ∩_k Ω_k is not empty by condition (B).' This implication is false in general: in Ξ = R, Ω_k = (0,1/k) are nonempty, open, nested, and have compact closures, yet ∩_k Ω_k = ∅. The proof never arranges the stronger inclusion cl(Ω_{k+1}) ⊂ Ω_k, which would allow Cantor's intersection theorem. The inductive step only produces Ω_{k+1} ⊂ U_w ∩ Ω_k with compact closure; it does not control the closure. Since every application (Theorems 2 and 3, and hence Theorem 1) obtains its limsup points as elements of ∩_k Ω_k, this gap is load-bearing: without a repair, the claimed existence of even one x0 with limsup f = c is not established. The gap is likely repairable: because the spaces in the applications are locally compact and Ω_k is open, one may choose U_w with cl(U_w) ⊂ Ω_k before applying local connectedness via (X_n), and then choose Ω_{k+1} inside U_w to get cl(Ω_{k+1}) ⊂ Ω_k. But the written proof does not do this.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for max(m,n)>1 and arbitrary norms on R^m and R^n, the Dirichlet spectrum D_{m,n} equals the full interval [0,Delta], and that the set of matrices attaining any prescribed limsup value is uncountable and dense in M_{m,n}. It also proves interval results for weighted psi-Dirichlet spectra and the associated uniform exponent (Theorems 2-3), formulates a general topological criterion (Theorem 5), and gives a negative result in dimension two (Theorem 4). The proofs combine a Khintchine-type topological construction with upper-uniformity estimates along rational affine subspaces and an equidistribution result of Kleinbock-Weiss.","tokens_in":21521,"tokens_out":16467,"duration_ms":173821,"significance":"If the gaps described below are repaired, this is a strong and significant paper. It resolves a natural generalization of Akhunzhanov-Shatskov, removes norm restrictions, and unifies several recent partial results under one topological mechanism. The verification of local connectedness via affine lines and planes and the parameter-free upper-uniformity bounds are explicit and careful, and the dependence on external results is clearly isolated: the equidistribution theorem of Kleinbock-Weiss and Mahler compactness are prior results, not derived from the paper's conclusion. The topological theorem is also likely to be useful beyond the present application.","major_comments":[{"comment":"The passage after condition (E) states: 'Omega_k is non-empty and open with compact closure for all k, hence the intersection cap_k Omega_k is not empty by condition (B).' This implication is false: a nested sequence of nonempty open sets with compact closures need not have nonempty intersection (for example (0,1/k) in R). The induction only guarantees Omega_{k+1} subset U_w cap Omega_k, not cl(Omega_{k+1}) subset Omega_k, so the existence of the point x_0 is not established. Since Theorems 2, 3 and 1 obtain their limsup points as elements of this intersection, this is a load-bearing gap. The repair is standard: in the locally compact setting one may first choose U_w with compact closure contained in Omega_k, then choose Omega_{k+1} with cl(Omega_{k+1}) subset Omega_k, making the closures a nested family of compact sets to which Cantor's theorem applies. The written proof should be amended to state and use this stronger inclusion.","section":"Section 4, proof of Theorem 5"},{"comment":"The statement of Theorem 2 includes the value c=0, but the proof only covers the interior values via Theorem 5 (which requires 0<c<b) and the value b via the full-measure equidistribution argument. No argument is supplied for c=0. For the special case E_epsilon=K_epsilon(||.||) this follows from rational matrices, and for an arbitrary continuous decreasing exhaustion it can be obtained similarly once one observes that a lattice Lambda_Theta with rational Theta contains a vector whose g_t-image tends to 0, forcing f(Theta,t) to tend to 0; however this is not stated in the proof. The endpoint c=0 therefore needs an explicit justification in the text.","section":"Section 5.1, proof of Theorem 2"},{"comment":"In the 'uncountably many' part of the proof of Theorem 5, the induction step at level k chooses U_w avoiding v_{k+1}, so it yields v_{k+1} notin Omega_{k+1}; it never excludes v_0 from Omega_0. The sentence 'repeating this argument at each step, we get that v_k notin Omega_k for all k' therefore does not follow for k=0. One can repair this by requiring at the base step that Omega_0 subset V setminus {v_0} (which is possible in the non-isolated spaces used in the applications, and otherwise the statement needs a separate discussion), and then the subsequent steps exclude v_1, v_2, and so on; this repair needs to be written explicitly.","section":"Section 4, uncountability part of the proof of Theorem 5"}],"minor_comments":[{"comment":"The definition of the lines X_{i,z} in the proof of Theorem 3 should use z in Q^m consistently; the text writes z in Z^m in the enumeration, but the local-connectedness argument from Section 5.1 requires the rational translates.","section":"Section 5.2, Case 1"},{"comment":"The notation 'open set V in M_{m,n}' should be 'open set V subset M_{m,n}' (and similarly in the corollaries); as written it suggests V is a matrix rather than a set.","section":"Theorems 1-3"},{"comment":"The manuscript contains numerous OCR-style typographical errors, including 'co rresponding', 'define', repeated bold words, and 'disjoints sets'; a careful proofreading pass is needed.","section":"Throughout"},{"comment":"The full-measure conclusion is stated for each fixed c<b; to obtain a single full-measure set on which limsup f >= b one should intersect over a countable sequence of c-values approaching b. This is standard but should be stated.","section":"Section 5.1"}],"recommendation":"major_revision","confidential_remarks":"The central result is very likely correct and the main topological gap is localized and repairable. The missing endpoint argument for c=0 and the base step of the uncountability argument are also local. I recommend major revision rather than rejection because the structural argument is sound and the stated claims are within reach after the repairs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you care about the Dirichlet spectrum. The paper claims the full interval result for all (m,n) ≠ (1,1) and arbitrary norms, subsuming a pile of partial results. That is a genuine advance, and the extended-spectrum framework (Theorem 2) plus the negative d=2 result (Theorem 4) are nice contributions. The topological theorem (Theorem 5) is a clean way to get uncountably many points at every level, and the applications to uniform exponents (Theorem 3 and Corollary 1) are meaningful.\n\nThe proof strategy is mostly honest: no fitted parameters, cites Kleinbock–Weiss equidistribution as an external input, and the case analysis for n=1 versus n≥2 is detailed. The upper uniformity argument, which is the technical heart, checks out as far as I can see.\n\nThe soft spot is real and load-bearing. In the proof of Theorem 5, after constructing nested nonempty open sets Ω_k with compact closures, the paper says 'hence the intersection is nonempty by condition (B).' That is just false: Ω_k=(0,1/k) in R is a counterexample. What is needed is cl(Ω_{k+1}) ⊂ Ω_k; the inductive step only gives Ω_{k+1} ⊂ U_w ∩ Ω_k, with no closure control. Since the point x0 in the intersection is what produces limsup f = c, Theorem 5 as written is unproved, and Theorems 1–3 inherit the gap.\n\nThat said, I believe it's repairable. The spaces in play are locally compact, Ω_k is open, and one can shrink U_w so that cl(U_w) ⊂ Ω_k before applying local connectedness, then choose Ω_{k+1} inside U_w. The uncountability argument also needs the same fix. As written, the paper needs a revised proof of Theorem 5. The mathematics around it looks strong enough that I expect the repair to be straightforward, but it is not a purely cosmetic issue.\n\nWho is this for? People working in Diophantine approximation and homogeneous flows. A serious referee should get it; it deserves review, but the referee should insist on the closure-nesting repair and a re-check of the induction.","headline":"Strong result, likely patchable gap: the proof of Theorem 5 asserts that nested nonempty open sets with compact closures have nonempty intersection, which is false, and the rest of the paper leans on it.","tokens_in":22045,"tokens_out":2154,"would_cite":true,"duration_ms":20296,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J13","11J83","37A17","11K60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every $(m,n)$ with $\\max(m,n)>1$ and every choice of norms on $\\mathbb{R}^m$ and $\\mathbb{R}^n$, the Dirichlet spectrum is the full interval $[0,\\Delta]$, with every level attained on an uncountable dense set.","keywords":["Dirichlet spectrum","Diophantine approximation","arbitrary norms","uniform exponent","unimodular lattices","homogeneous diagonal flows","irrationality measure","topological interval theorem"],"falsifier":"Take a bounded positive-measure set $B\\subset M_{m,n}$ and a compactly supported continuous function $F$ supported inside one compact set $E_c$ of the exhaustion, and check whether the averaged integral $\\mathrm{vol}(B)^{-1}\\int_B F(g_{t,\\vec\\alpha,\\vec\\beta}\\Lambda_\\Theta)\\,d\\mathrm{vol}(\\Theta)$ approaches the Haar-Siegel average for large $t$; if the equidistribution statement of Theorem 6 fails for some weights or norms, the dense starting set used in the induction does not exist and the interval conclusion is unsupported for that case. A second check is to verify that in an explicit application of Theorem 5 the constructed sets $\\Omega_k$ have closures whose intersection is nonempty.","tokens_in":21008,"feed_emoji":"🎯","tokens_out":11738,"duration_ms":101002,"temperature":0.7,"pith_summary":"This paper establishes that the Dirichlet spectrum is an interval in every nontrivial dimension: for any $m \\times n$ real matrices with $\\max(m,n)>1$, and for any two norms on $\\mathbb{R}^m$ and $\\mathbb{R}^n$, every value between $0$ and the best Dirichlet constant $\\Delta$ occurs as $\\limsup_{t\\to\\infty} \\chi(\\Theta,t)$ for some matrix $\\Theta$, and the set of matrices realizing each value is uncountable and dense. This completes a line of results that had been proved only for special pairs of norms and dimensions. The proof isolates a purely topological mechanism: a continuous Diophantine function that is uniformly small on a dense family of rational affine subspaces and has a dense set of points with large limsup must realize every intermediate limsup. The same mechanism yields interval spectra for the lattice-theoretic extended Dirichlet constant, for weighted $\\psi$-Dirichlet spectra, and for prescribed uniform exponents, and it explains why the $(1,1)$ case and the two-dimensional lattice version genuinely fail to be intervals.","feed_headline":"Dirichlet spectrum fills the whole interval [0, Δ]","feed_subtitle":"Beyond the one-by-one case, each approximation rate from 0 to the Dirichlet bound occurs on a dense uncountable set.","key_machinery":"The load-bearing object is the topological interval theorem, Theorem 5. It says that if $\\Xi$ is a Hausdorff space, $f:\\Xi\\times(0,\\infty)\\to[0,\\infty]$ is continuous, and there is a sequence of subsets $(X_n)$ such that $\\Xi$ is locally connected via $(X_n)$ and the $X_n$ satisfy upper uniformity, meaning $f(x,t)\\to 0$ uniformly on compact sets as $t\\to\\infty$ for $x\\in X_n$, then whenever the set of points with $\\limsup_{t\\to\\infty} f(x,t)\\ge b$ is dense, every level $c\\in(0,b)$ is realized by uncountably many points. The construction threads a point through shrinking connected pieces $\\Omega_k$ that alternately force $f$ below $c$ on long time intervals and above $c-\\varepsilon_k$ at specified moments. In the applications, $\\Xi$ is the matrix space $M_{m,n}$, $f$ is either the continuous function $\\lambda_{\\Theta,\\psi}(t)$ or the return function $\\inf\\{\\varepsilon: g_{t,\\vec\\alpha,\\vec\\beta}\\Lambda_\\Theta\\notin E_\\varepsilon\\}$, and the sets $X_n$ are rational affine lines or planes, whose upper uniformity is verified using Dirichlet's theorem and explicit vector estimates.","core_discovery":"The central discovery is that the Dirichlet spectrum is an interval $D_{m,n}=[0,\\Delta]$ for all $m,n$ with $\\max(m,n)>1$ and arbitrary norms, where $\\Delta$ is the minimal constant in the higher-dimensional Dirichlet theorem. Moreover, the level sets $\\{\\Theta\\in M_{m,n}: \\limsup_{t\\to\\infty}\\chi(\\Theta,t)=c\\}$ are uncountable and dense for each $c\\in[0,\\Delta]$. The same statement is proved for the extended Dirichlet constant on unimodular lattices in dimension $d\\ge 3$ with arbitrary weights and arbitrary continuous compact exhaustions, for the $\\psi$-Dirichlet spectrum with $\\psi(t)=o(t^{-1})$, and for the uniform exponent $\\hat\\omega(\\Theta)$, while a separate theorem shows the analogous lattice spectrum in dimension $d=2$ has a gap, so the interval phenomenon is genuinely a high-dimensional one.","pith_inferences":["Implicit in the paper: the same topological theorem should produce interval spectra for other continuous Diophantine quantities with a rational-affine upper-uniform family and a dense high-limsup set, such as weighted approximations with time-dependent weights.","Implicit in the paper: the obstruction in the $(1,1)$ case is presented as topological, since the auxiliary sets collapse to points, which suggests a geometric criterion for interval spectra rather than an arithmetic one.","A testable extension: one can numerically probe a single level $c$ near $\\Delta$ by starting from a rational matrix on one of the affine lines used in the proof and adding a tiny perturbation; the proof predicts the limsup stays exactly $c$ for uncountably many nearby matrices, so stability of the limsup under such perturbations would corroborate the dense-level claim.","Implicit in the paper: the $d=2$ gap and the $(1,1)$ failure suggest the natural next question is the sharp size and location of the gap for specific exhaustions, and whether the gap persists for all proper metric exhaustions in $d=2$."],"forward_implications":["For every $c\\in[0,\\Delta]$, uncountably many matrices in any open set have $\\limsup_{t\\to\\infty}\\chi(\\Theta,t)=c$, so the spectrum is not just an interval but has every level dense.","For any positive continuous decreasing $\\psi(t)=o(t^{-1})$ satisfying the extra decay condition when $n=1$, the $\\psi$-Dirichlet spectrum is $[0,\\infty]$, giving prescribed limsup values for the continuous approximation function $\\lambda_{\\Theta,\\psi}$.","For every uniform exponent $\\gamma>n/m$, with $\\gamma<1$ when $n=1$, there are uncountably many dense matrices with $\\hat\\omega(\\Theta)=\\gamma$ and with $\\limsup_{t\\to\\infty}\\chi_\\gamma(\\Theta,t)$ equal to any prescribed $c\\in[0,\\infty]$.","In dimension $d\\ge 3$, for arbitrary weights and any continuous decreasing compact exhaustion of the lattice space, the extended Dirichlet spectrum is $[0,b]$, which answers previously open questions about metric-ball versions of the spectrum.","For $d=2$, the extended spectrum always contains a gap $(0,r)$, so the full-interval phenomenon fails there, marking the dimensional threshold."],"supporting_citations":[{"why":"Origin of the topological method: Theorem 5 is a modification of Khintchine's construction of non-obvious singular vectors.","marker":"[Khi26]"},{"why":"Supplies the intermediate version of the topological theorem whose inductive pattern Theorem 5 adapts.","marker":"[Wei04]"},{"why":"Provides the equidistribution theorem, cited as Theorem 6, that proves the full-measure dense set with maximal limsup.","marker":"[KW08]"},{"why":"Contributes the weighted analog of the Davenport-Schmidt argument used to identify the almost-everywhere limsup value.","marker":"[KR21]"},{"why":"Defined the Dirichlet spectrum and proved the $(2,1)$ Euclidean interval case that this paper generalizes.","marker":"[AS13]"},{"why":"Established the almost-everywhere limsup equals Delta result in the one-row and one-column max-norm cases, the base typical-value fact.","marker":"[DS70]"}],"fun_headline_variants":["Dirichlet spectrum is a full interval for all dimensions","Every rate from 0 to Δ occurs in Dirichlet spectrum","No gaps: Dirichlet spectrum covers [0, Δ] completely","Interval result: spectrum is continuous for all norms","Dirichlet spectrum fills [0, Δ] for every non-trivial case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the equidistribution theorem cited as [KW08] applies to the weighted diagonal flow for every bounded set used in the induction, because it alone provides the dense full-measure set of matrices with maximal limsup from which the level-by-level construction starts; the proof of Theorem 5 also needs a repair in the nesting of open sets, since compact-closure nesting is asserted rather than derived.","fun_headline_variants_meta":{"raw":{"variants":["Dirichlet spectrum is a full interval for all dimensions","Every rate from 0 to Δ occurs in Dirichlet spectrum","No gaps: Dirichlet spectrum covers [0, Δ] completely","Interval result: spectrum is continuous for all norms","Dirichlet spectrum fills [0, Δ] for every non-trivial case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3391,"prompt_tokens":839,"completion_tokens":2552,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":455,"completion_tokens_details":{"reasoning_tokens":2467}},"tokens_in":455,"tokens_out":2552,"duration_ms":18787,"temperature":1.0,"reasoning_tokens":2467,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:20:25.528870+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a bounded positive-measure set $B\\subset M_{m,n}$ and a compactly supported continuous function $F$ supported inside one compact set $E_c$ of the exhaustion, and check whether the averaged integral $\\mathrm{vol}(B)^{-1}\\int_B F(g_{t,\\vec\\alpha,\\vec\\beta}\\Lambda_\\Theta)\\,d\\mathrm{vol}(\\Theta)$ approaches the Haar-Siegel average for large $t$; if the equidistribution statement of Theorem 6 fails for some weights or norms, the dense starting set used in the induction does not exist and the interval conclusion is unsupported for that case. A second check is to verify that in an explicit application of Theorem 5 the constructed sets $\\Omega_k$ have closures whose intersection is nonempty.","supporting_citations":[],"review_version":1}