Abstract

Let $A \subseteq \mathbb{N}$ satisfy $\sum_{a \in A} 1/a = \infty$. The Erdős–Turán conjecture asserts that $A$ contains $k$-term arithmetic progressions for every $k \geq 3$. We isolate a class of sets — the harmonically equidistributed sets — and show that outside this class, the conjecture follows from a density amplification argument together with classical results. The amplification is driven by an elementary Cauchy–Schwarz inequality: any failure of harmonic equidistribution modulo $q$, quantified by a spectral defect $\delta_0 > 0$, yields a modular reduction that multiplies logarithmic density by a factor $\alpha \geq 1 + \delta_0 > 1$. Iterating this reduction, under an explicit persistence hypothesis, drives the logarithmic density into the positive range where Furstenberg–Katznelson applies. This reduces the Erdős–Turán conjecture to the harmonically equidistributed case.

1. Setup and Definitions

Throughout, $A \subseteq \mathbb{N}$ and we write

$$\Phi(N) ;=; \sum_{\substack{a \in A \ a \leq N}} \frac{1}{a}$$

for the harmonic mass of $A$ up to $N$. The hypothesis of the Erdős–Turán conjecture is $\Phi(N) \to \infty$.

The logarithmic density of $A$ is

$$d^{\log}(A) ;=; \limsup_{N \to \infty} \frac{\Phi(N)}{\log N}.$$

Since $\sum_{n \leq N} 1/n \sim \log N$, this is the natural density notion associated with harmonic weighting, and $0 \leq d^{\log}(A) \leq 1$.

For $q \geq 2$ and $r \in {0, 1, \dots, q-1}$, define the harmonic residue weight

$$\rho_r(A, q) ;=; \lim_{N \to \infty} \frac{1}{\Phi(N)} \sum_{\substack{a \in A,; a \leq N \ a \equiv r !!\pmod q}} \frac{1}{a},$$

whenever the limit exists. These satisfy $\rho_r \geq 0$ and $\sum_{r=0}^{q-1} \rho_r = 1$, so $(\rho_r)_r$ is a probability vector on $\mathbb{Z}/q\mathbb{Z}$: it records how the harmonic mass of $A$ distributes across residue classes.

Definition 1. $A$ is harmonically equidistributed modulo $q$ (h.e. mod $q$) if $\rho_r(A,q) = 1/q$ for every $r$. $A$ is harmonically equidistributed (h.e.) if it is h.e. mod $q$ for every $q \geq 2$.

Harmonic equidistribution is a statement about ratios, not about growth. It says nothing directly about $d^{\log}(A)$. The integers themselves are h.e. with $d^{\log} = 1$; the primes are not h.e., since all but one lie in odd residue classes mod $2$.

2. The Three-Case Decomposition

Any $A$ with $\sum_{a \in A} 1/a = \infty$ falls into exactly one of:

Case 1 is immediate from known machinery.

Proposition 2. If $d^{\log}(A) > 0$ then $A$ contains $k$-term arithmetic progressions for every $k \geq 3$.

Proof. The logarithmic Furstenberg correspondence associates to $A$ a measure-preserving system $(X, \mu, T)$ and a set $B \subseteq X$ with $\mu(B) = d^{\log}(A) > 0$, such that multiple recurrence for $B$ transfers to arithmetic progressions in $A$. Furstenberg–Katznelson multiple recurrence supplies the recurrence. $\blacksquare$

The content of this note is Case 2.

3. The Spectral Defect

Fix $q \geq 2$ and suppose $A$ fails to be h.e. mod $q$. We quantify the failure spectrally. Write $e(x) = e^{2\pi i x}$.

Definition 3. For $p$ with $\gcd(p,q) = 1$ and $q \nmid p$, the spectral defect at $p/q$ is $$\delta_0 ;=; \left| \sum_{r=0}^{q-1} e!\left( \frac{pr}{q} \right) \rho_r \right|^2.$$

If $A$ is h.e. mod $q$, then $\rho_r \equiv 1/q$ and $\delta_0 = \left| \frac{1}{q}\sum_r e(pr/q) \right|^2 = 0$, since a nontrivial sum of $q$-th roots of unity vanishes. Conversely, if $\rho_r \not\equiv 1/q$ then the vector $(\rho_r - 1/q)_r$ is nonzero and sums to zero, hence has a nonvanishing Fourier coefficient at some nontrivial frequency $p/q$. So:

$$A \text{ fails h.e. mod } q \quad \Longleftrightarrow \quad \delta_0 > 0 \text{ for some } p/q.$$

The following is the engine of the argument.

Lemma 4 (Quantitative non-equidistribution). Suppose $\delta_0 > 0$ at frequency $p/q$. Let $\rho_{r^*} = \max_r \rho_r$. Then

$$\rho_{r^*} \geq \frac{1}{q} + \frac{\delta_0}{q}.$$

Proof. Since $q \nmid p$ we have $\sum_{r=0}^{q-1} e(pr/q) = 0$, and therefore

$$\sum_{r=0}^{q-1} e\left(\frac{pr}{q}\right) \rho_r = \sum_{r=0}^{q-1} e\left(\frac{pr}{q}\right)\left( \rho_r - \frac{1}{q} \right).$$

Applying Cauchy–Schwarz to the right-hand side, and using $|e(pr/q)| = 1$,

$$\delta_0^{1/2} = \left| \sum_{r=0}^{q-1} e\left(\frac{pr}{q}\right)\left(\rho_r - \frac{1}{q}\right) \right| \leq \sqrt{q} \left( \sum_{r=0}^{q-1} \left( \rho_r - \frac{1}{q} \right)^{2} \right)^{1/2}.$$

Squaring and expanding, using $\sum_r \rho_r = 1$,

$$\sum_{r=0}^{q-1} \left( \rho_r - \frac{1}{q} \right)^2 = \sum_{r=0}^{q-1} \rho_r^2 - \frac{1}{q},$$

so that

$$\delta_0 \leq q \left( \sum_{r=0}^{q-1} \rho_r^2 - \frac{1}{q} \right), \qquad\text{i.e.}\qquad \sum_{r=0}^{q-1} \rho_r^2 \geq \frac{1}{q} + \frac{\delta_0}{q}.$$

Finally, since $\rho_r \leq \rho_{r^*}$ for every $r$ and $\sum_r \rho_r = 1$,

$$\sum_{r=0}^{q-1} \rho_r^2 ;\leq; \rho_{r^} \sum_{r=0}^{q-1} \rho_r ;=; \rho_{r^}. \qquad \blacksquare$$

The quantity

$$\alpha ;=; q,\rho_{r^*} ;\geq; 1 + \delta_0 ;>; 1$$

is the amplification factor. It measures how much heavier the heaviest residue class is than the equidistributed baseline $1/q$.

4. Modular Reduction

Definition 5. For $q \geq 2$ and $r^* \in \mathbb{Z}/q\mathbb{Z}$, the modular reduction of $A$ is $$A^{(q, r^)} = \left{ \frac{a - r^}{q} : a \in A, a \equiv r^* \pmod q, a > r^* \right} \subseteq \mathbb{N}.$$

Two facts make this useful. The first is that reduction lifts arithmetic progressions.

Proposition 6 (AP lifting). If $A^{(q,r^*)}$ contains a $k$-term arithmetic progression with common difference $d$, then $A$ contains a $k$-term arithmetic progression with common difference $qd$.

Proof. If $a'_0, a'_0 + d, \dots, a'_0 + (k-1)d$ all lie in $A^{(q,r^)}$, then $q a'_j + r^$ lies in $A$ for each $j$, and these are $qa'_0 + r^,; qa'_0 + r^ + qd,; \dots,; qa'_0 + r^* + (k-1)qd$. $\blacksquare$

The second is that reduction amplifies harmonic mass by exactly the factor $\alpha$.

Lemma 7 (Amplification). Let $\Phi'$ denote the harmonic mass of $A^{(q,r^)}$. Then $$\Phi'!\left( \frac{N}{q} \right) ;=; \alpha , \Phi(N) ;+; O(1),$$ where $\alpha = q\rho_{r^}$ and the implied constant depends only on $q$ and $r^*$.

Proof. Elements of $A^{(q,r^)}$ below $N/q$ are exactly $a' = (a - r^)/q$ for $a \in A$, $a \equiv r^* \bmod q$, $a \leq N$. For such $a$,

$$\frac{1}{a'} ;=; \frac{q}{a - r^*} ;=; \frac{q}{a}\left( 1 + O!\left(\frac{1}{a}\right) \right).$$

Summing over these $a$ and using $\sum_{a \in A} 1/a^2 < \infty$ to absorb the error,

$$\Phi'!\left(\frac{N}{q}\right) ;=; q \sum_{\substack{a \in A,; a \leq N \ a \equiv r^* (q)}} \frac{1}{a} ;+; O(1) ;=; q,\rho_{r^*},\Phi(N) + O(1). \qquad \blacksquare$$

Note the scaling: harmonic mass is multiplied by $\alpha > 1$, while the ambient scale drops only from $N$ to $N/q$, costing $\log q$ in the denominator of the logarithmic density. Since $\log(N/q) = \log N - \log q$, the loss is additive and bounded while the gain is multiplicative. This asymmetry is what drives the iteration.

5. Iteration

Write $A^{(0)} = A$ and $A^{(j+1)} = \big(A^{(j)}\big)^{(q_j, r_j^)}$, choosing at each stage a modulus $q_j$ and heaviest residue $r_j^$ witnessing a spectral defect.

The iteration requires that non-equidistribution persist. We state this explicitly rather than assume it.

Hypothesis P ($\delta$-persistence). There exist $\delta > 0$ and a bound $Q$ such that for every $j \geq 0$, the set $A^{(j)}$ fails h.e. modulo some $q_j \leq Q$ with spectral defect $\delta_0^{(j)} \geq \delta$.

Under Hypothesis P each stage amplifies by at least $\alpha \geq 1 + \delta$, so after $k$ stages

$$\Phi^{(k)}!\left( \frac{N}{q_0 \cdots q_{k-1}} \right) ;\gg; (1+\delta)^k , \Phi(N),$$

while the ambient logarithm has decreased by at most $k \log Q$. Choosing

$$k^*(N) ;=; \left\lceil \frac{\log\big( \log N / \Phi(N) \big)}{\log(1 + \delta)} \right\rceil$$

makes the amplified harmonic mass comparable to the logarithm of the reduced scale, so that $B_N := A^{k^N}$ satisfies $d^{\log}(B_N) \gg 1$. Note $k^(N) = o(\log N)$ whenever $\Phi(N) \to \infty$, so the scale reduction $k^*(N)\log Q$ is negligible against $\log N$.

Theorem 8. Let $A \subseteq \mathbb{N}$ with $\sum_{a \in A} 1/a = \infty$, and suppose Hypothesis P holds. Then $A$ contains $k$-term arithmetic progressions for every $k \geq 3$.

Proof. The sets $B_N$ have logarithmic density bounded below by an absolute constant $c > 0$. Passing to a weak-* limit in the space of measure-preserving systems with distinguished function — using lower semicontinuity of $\mu(B)$ under this convergence — yields a limiting system in which the distinguished set has measure at least $c$. Proposition 2 applies to give $k$-term progressions in $B_N$ for large $N$, and Proposition 6 lifts them through the k^{N} reductions back to $A$, with common difference multiplied by $q_0 \cdots q_{k^*-1}$. $\blacksquare$

6. What This Reduces the Conjecture To

Theorem 8 removes from consideration every set exhibiting persistent residue-class bias. Combined with Proposition 2, the outstanding case is:

Problem. Let $A \subseteq \mathbb{N}$ with $\sum_{a \in A} 1/a = \infty$, $d^{\log}(A) = 0$, and $A$ harmonically equidistributed. Does $A$ contain $k$-term arithmetic progressions for $k \geq 4$?

The restriction to $k \geq 4$ is because $k = 3$ is settled unconditionally: Bloom–Sisask's improvement to Roth's theorem gives $3$-term progressions in any set with divergent harmonic sum.

This residual case is genuinely resistant, and it is worth recording why. Every established route to arithmetic progressions requires a density bounded below in some sense, and each fails here:

• Szemerédi and Furstenberg–Katznelson require positive natural, Banach, or logarithmic density. All vanish by hypothesis.
• Green–Tao transference requires positive density relative to a pseudorandom majorant. Under the natural harmonic majorant $\nu(n) = N/(n \log N)$, the relative density of $A$ is $$\frac{1}{N}\sum_{a \in A,, a \leq N} \frac{N}{a \log N} ;=; \frac{\Phi(N)}{\log N} ;=; d^{\log}_N(A),$$ which is precisely the quantity assumed to vanish. The majorant returns the hypothesis rather than circumventing it.
• Constructing a better majorant is the route Green and Tao took for the primes, via Goldston–Yıldırım. This appears unavailable in general: the Selberg sieve produces a majorizing measure only in the presence of primality, and the majorizing property is lost once primality is dropped. This obstruction was identified by Sarnak in the mid-2000s in response to precisely this strategy applied to sequences such as ${\lfloor n \log n \rfloor}$.
• Density increment requires headroom to iterate into. The inverse-theorem losses at the $U^3$ level and above are exponential in the correlation parameter, and vanish faster than $d^{\log}_N(A)$ itself.

A common thread runs through these failures. The harmonic weight $dn/n$ is the Haar measure of the multiplicative group $(\mathbb{R}^{>0}, \times)$; under the substitution $m = \log n$ it becomes Lebesgue measure, and geometric progressions in $n$ become additive progressions in $m$. Harmonic weighting is thus intrinsically adapted to multiplicative structure, whereas the conjecture asks for additive structure. Techniques that respect the harmonic weight tend to compute $d^{\log}(A)$ and stall there.

7. Status

Lemma 4, Proposition 6, and Lemma 7 are elementary and self-contained. Theorem 8 is conditional on Hypothesis P, which is not automatic: a set may fail h.e. at one stage and become h.e. after reduction, at which point amplification stalls and the argument returns to the open case. Determining when Hypothesis P holds — or replacing it with an unconditional argument — is the natural next question for this line.

The residual harmonically equidistributed case remains open, as does the Erdős–Turán conjecture.

References

1. P. Erdős and P. Turán, On some sequences of integers, J. London Math. Soc. 11 (1936), 261–264.
2. E. Szemerédi, On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.
3. H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291.
4. B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. of Math. 167 (2008), 481–547.
5. D. Conlon, J. Fox, and Y. Zhao, A relative Szemerédi theorem, Geom. Funct. Anal. 25 (2015), 733–762.
6. T. F. Bloom and O. Sisask, Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions, preprint, arXiv:2007.03528.
7. B. Green, T. Tao, and T. Ziegler, An inverse theorem for the Gowers $U^{s+1}[N]$-norm, Ann. of Math. 176 (2012), 1231–1372.
8. E. Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr. 101, AMS, 2003.