arXiv:2610.27497v1

On a Notion of Density Related to Restricted Diophantine Approximation

math.NT11J0411B0511J25

Paper metadata

Original submission: October 20, 2026
Claimed MSC: 11J04, 11B05, 11J25
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: 15 pages. Comments welcome. This paper significantly extends the results of arXiv:2606.02620 by the same author

Abstract

We introduce notion of density for a subset of the natural numbers called the Diophantine density. First, we demonstrate that the Diophantine density is an abstract density in the sense of Di Nasso and Jin. Then we establish upper and lower bounds for the Diophantine density of a set and its complement respectively. We also exhibit a construction which allows us to find a set whose density can be any value between zero and one. Finally, we prove a restricted-denominator analog of Dirichlet's approximation theorem for sets of non-zero Diophantine density.

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Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 23, 2026

01Statements3 reported findingsNo statement issues found

The abstract-density properties, counting-function bounds, prescribed-density construction, and restricted approximation theorem are supported. One local domain restriction is needed in the auxiliary finite-minimum lemma; it does not affect the central theorems.

Theorem 3; Propositions 1–6No issue found

The density is normalized, monotone, and neither translation invariant nor homogeneous

PDF pp. 2–5 · arXiv:2610.27497v1

For a nonempty set, exponent zero is admissible once a fixed member lies below the denominator. Invertibility modulo $q$ gives $\delta(\mathbb N)=1$ , while a finite set cannot make all the distances to an irrational limit tend to zero. Inclusion gives monotonicity. Dirichlet approximation of $mp/q$ gives $\delta(m\mathbb N)=1$ . For the translated progression, the coprime fractions $n/(2mn+1)$ keep every permitted distance above $1/4$ , proving the asserted failure of translation invariance.

Theorems 4–6No issue found

Counting bounds and the prescribed-density construction have the stated exponents

PDF pp. 6–10 · arXiv:2610.27497v1

Removing $A(q)$ residues bounds the surviving distance by $(A(q)+1)/q$ whenever a residue survives. In the only nontrivial application, $A(q)=o(q)$ guarantees survival. Conversely, prime denominators give $q-1\leq 2CA(q)q^{1-\delta}$ , and Bertrand's postulate extends this estimate to all large integers. For $A_x$ , Dirichlet's theorem on a block gives the lower exponent $x$ , while splitting the block-counting sum at $\lfloor\log_2 n\rfloor$ gives $A_x(n)\ll_x n^x$ , hence the matching upper exponent.

Theorem 7; Corollary 1; Propositions 7–9No issue found

The restricted irrationality and power-sequence consequences follow

PDF pp. 10–13 · arXiv:2610.27497v1

Choose $\delta<\delta'<\delta(A)$ and apply the defining estimate to continued-fraction convergents. The resulting error is at most $(C+1)v^{-1-\delta'}$ ; irrationality forces infinitely many unbounded permitted denominators, so the constant is absorbed by $v^{\delta'-\delta}$ . No coprimality of the final pair is claimed. The count $S_\tau(n)\asymp n^{1/\tau}$ gives the stated upper density and complement bounds. Substitution of denominators below $\sqrt q$ in Zaharescu's estimate gives the square-set lower exponent $2/7$ .

02Proofs3 reported findingsNo proof issues found

The residue-counting, prime-denominator, block-construction, and convergent-transfer arguments support the central conclusions. The auxiliary Lemma 1 needs a local nonempty-domain qualification, recorded below without lowering the substantive proof status.

Lemma 1Suspected minor formal issue

Restrict the finite minimum to cases where its indexing set is nonempty

PDF p. 6 · Lemma 1 and its proof · arXiv:2610.27497v1

The lemma is quantified over every $A\subseteq\mathbb N$ and every coprime pair, but the minimum is undefined when $\{v\in\mathbb N\setminus A:v<q\}$ is empty; $A=\mathbb N$ is an example. Add the hypothesis $A(q)<q-1$ . Under that hypothesis the finite residue argument proves the displayed estimate. Theorem 4 uses the lemma only after reducing to $A(q)\ll q^\tau$ with $\tau<1$ , which supplies this hypothesis for all sufficiently large $q$ ; its conclusion and every later application are unaffected.

Theorems 4 and 5No issue found in the reviewed proof

The limiting exponents and quantifiers are consistent

PDF pp. 6–8 · arXiv:2610.27497v1

The infimum of admissible counting exponents equals the upper logarithmic growth exponent by the two direct logarithmic inequalities. For the upper density bound, every exponent strictly below $\delta(A)$ is admissible because admissibility is downward closed. The prime-residue union estimate is uniform in the numerator, and monotonicity of the counting function transfers its lower bound from a prime in $[n/2,n]$ to $n$ . The zero-density case follows immediately from nonnegativity.

Theorems 6 and 7No issue found in the reviewed proof

The block tail and the absorption of constants close the arguments

PDF pp. 9–12 · arXiv:2610.27497v1

The tail estimate uses $d_k\geq 2^{(1-x)k}/2$ , so its geometric series converges exactly for $x<1$ ; the initial block sum converges after factoring $n^x$ exactly for $x>0$ . In the approximation proof, $v<q_n$ and $1+\delta'<2$ give the correct error comparison. Any bounded infinite subsequence of the $v$ 's would force $\|v\xi\|=0$ for a fixed positive integer $v$ , contradicting irrationality. This justifies passing to unbounded denominators and removing the fixed constant.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2610.27497v1
Authors listed: Chance Sanford
Audit date: October 23, 2026

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On a Notion of Density Related to Restricted Diophantine Approximation

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The density is normalized, monotone, and neither translation invariant nor homogeneous

Counting bounds and the prescribed-density construction have the stated exponents

The restricted irrationality and power-sequence consequences follow

Restrict the finite minimum to cases where its indexing set is nonempty

The limiting exponents and quantifiers are consistent

The block tail and the absorption of constants close the arguments

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