Mapping Incidences

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theorem_1_1: For a finite subset S of a characteristic-zero integral domain and a finite set L of nonzero elements of the ring Z[S], there is a sequence of primes of positive relative density for each of which some ring homomorphism from Z[S] to Z/pZ sends no element of L to zero.

theorem_2_3: For at most N points and at most N lines in D x D, where D is any characteristic-zero integral domain, the number of incidences is at most c N^(3/2-delta) for positive absolute constants c and delta, those of the finite-field bound the paper quotes as its Theorem 2.1.

theorem_3_2: For every finite subset A of a characteristic-zero integral domain, the larger of the sumset and the product set has size at least C|A|^(14/13)(log|A|)^alpha, for positive absolute constants C and alpha that the paper says are those of the Katz-Shen bound for Z/pZ it quotes as its Theorem 3.1.

theorem_4_2: For a finite subset A of SL_2(D), with D a characteristic-zero integral domain, if the group generated by A is infinite and not metabelian, then |AAA| > c|A|^(1+delta) for absolute constants c > 0 and delta > 0.

theorem_5_1: For every rho < 1 there is delta < 1 such that an n by n matrix with iid entries, each a finitely supported random variable in a characteristic-zero integral domain taking each value with probability at most rho, is singular with probability at most delta^n for all n large in terms of rho and the support size.

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Van H. Vu, Melanie Matchett Wood, Philip Matchett Wood, "Mapping Incidences," arXiv:0711.4407 (2007).

The copy read for this card is arXiv:0711.4407v2 of 15 April 2011, the current arXiv version: the arXiv v2 manuscript (stamp "arXiv:0711.4407v2 [math.CO] 15 Apr 2011" on p. 1, which carries the title "Mapping Incidences", the three authors and the abstract), 15 pages whose printed numbers equal the PDF page numbers, with a text layer. Provenance: downloaded from https://arxiv.org/pdf/0711.4407v2 on 2026-09-22; 211,592 bytes. The arXiv comments field says the paper was to appear in the Journal of the London Mathematical Society and that v2 dropped v1's Section 3 on the Erdős distance problem; the journal version, J. London Math. Soc. (2) 84 (2011), no. 2, 433--445, DOI 10.1112/jlms/jdr017 (Crossref record read), was not acquired and no version of record was compared. The locators below are the v2 text's section headings and numbered statements (Sections 1--8, Theorems 2.1, 2.3, 3.1 and 3.2, Lemma 5.4, Theorem 6.1, Lemma 7.1, Remark 7.2), with the printed page numbers, all checked on the printed pages: Section 1 runs pp. 1--3, Section 2 pp. 3--4, Section 3 pp. 4--5, Section 4 pp. 5--8, Section 5 pp. 9--10, Section 6 pp. 10--11, Section 7 pp. 11--13, and Section 8 with the references pp. 13--15. The arXiv record names arXiv's non-exclusive distribution license (arXiv:0711.4407), every other right reserved.

The text was read in full. Read status: claims checked for the results below; the proofs were read for their mechanism but were not independently verified.

The mapping theorem

Theorem 1.1 (p. 2; proved in Section 7, pp. 11--13) says: if $S$ is a finite subset of a characteristic-zero integral domain $D$, and $L$ is a finite set of nonzero elements of the subring $\mathbb Z[S]$, then there is an infinite sequence of primes of positive relative density such that, for every prime $p$ in the sequence, there is a unital ring homomorphism

$$\phi_p:\mathbb Z[S]\longrightarrow\mathbb Z/p\mathbb Z$$

with $\phi_p(\ell)\ne0$ for every $\ell\in L$. The theorem is qualitative: it does not bound the least usable prime in terms of $S$ and $L$ (the discussion after Theorem 1.1 on p. 2, and Remark 7.2 on p. 11).

The proof first specializes to an algebraic-number ring without killing $L$ (Lemma 7.1, p. 11), then uses the Frobenius Density Theorem (Theorem 6.1, p. 10) to choose a positive-density set of primes for which the required defining polynomial splits; the final quotient construction is in the proof of Theorem 1.1 in Section 7.

This preserves any prescribed finite zero/nonzero pattern of integer-coefficient polynomial expressions in elements of $S$. True polynomial equalities are preserved because $\phi_p$ is a ring homomorphism; putting each nonzero residual into $L$ prevents false equalities from becoming true. In particular, suitable choices of $L$ preserve:

• distinctness and hence cardinality, by protecting $s-t$ for distinct elements;
• additive incidences $a_1+a_2=a_3+a_4$ and multiplicative incidences $a_1a_2=a_3a_4$;
• point-line incidences $y=mx+b$ (Theorem 2.3, p. 3); and
• vanishing of determinants, in both directions, for all $n\times n$ matrices with entries in $S$, for a given $n$ (Lemma 5.4, p. 9, the step behind Theorem 5.1).

Theorem 2.3 applies this device to the finite-field incidence estimate of Theorem 2.1 (p. 3), which the paper quotes as Theorem 6.2 of Bourgain, Katz and Tao, valid for all $N\le q$ once later sum-product estimates replace theirs (Remark 2.2), obtaining the same $cN^{3/2-\delta}$ point-line incidence bound in every characteristic-zero integral domain. Its proof protects the differences needed to keep the point and line parameter sets distinct, while the equation $y=mx+b$ passes through the homomorphism.

Sum-product application

For a finite $A\subset D$, the proof of Theorem 3.2 (p. 5) takes $L$ to contain every nonzero expression of the forms

$$a_1-a_2,\qquad a_1+a_2-(a_3+a_4),\qquad a_1a_2-a_3a_4$$

with $a_i\in A$. It may then choose $p>|A|^2$ and obtains exactly

$$|\phi_p(A)|=|A|,\qquad |\phi_p(A)+\phi_p(A)|=|A+A|,\qquad |\phi_p(A)\phi_p(A)|=|AA|.$$

Applying the Katz--Shen finite-field estimate quoted as Theorem 3.1 (p. 4), which assumes $|A|<p^{1/2}$, therefore gives absolute $C,\alpha>0$ such that every finite subset of every characteristic-zero integral domain satisfies

$$C|A|^{14/13}(\log|A|)^\alpha \le \max\{|A+A|,|AA|\}.$$

More generally, the same argument transfers any finite-field lower bound whose hypotheses can be expressed by finitely many protected algebraic incidences; the availability of arbitrarily large usable primes handles conditions such as $|A|<p^{1/2}$.

Direction of transfer

The theorem maps a fixed finite characteristic-zero configuration to a finite field. It does not map a finite-field or real configuration into $\mathbb Q$ or $\mathbb Z$, and it does not realize the generated characteristic-zero ring inside either one. Thus, even if one maps each member of a family of real sum-product constructions to some finite field while preserving $|A|$, $|A+A|$, and $|AA|$, the output is only a family in varying finite fields.

This is decisive for constructions made in totally real number fields whose degrees grow with $|A|$: Theorem 1.1 chooses a separate prime and specialization for each finite input, but supplies no bounded-degree model, embedding, or specialization in $\mathbb Q$ or $\mathbb Z$. Consequently it cannot turn such a growing-degree real counterexample into an integer counterexample to Problem 52. Its relevance to the integer problem is instead the forward transfer of finite-field lower bounds, exemplified by the exponent $14/13$ above.

Results. Theorem 1.1 (p. 2, with Lemma 7.1 of p. 11 on its page); Theorem 2.3 (p. 3, with the quoted Theorem 2.1); Theorem 3.2 (p. 5, with the quoted Theorem 3.1); Theorem 4.2 (p. 6); Theorem 5.1 (p. 9, with Lemma 5.4 on its page).

Bears on. #52: Theorem 3.2 gives $\max(|A+A|,|AA|)\ge C|A|^{14/13}(\log|A|)^\alpha$ for finite sets of integers, by transferring a finite-field bound through Theorem 1.1. The exponent $14/13$ is far below the $2-\epsilon$ the problem asks about, and the paper does not compare it with exponents already known over the reals. The paper neither settles the problem nor transfers growing-degree real constructions into $\mathbb Q$ or $\mathbb Z$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.