Effective joint Fourier decay at independent Pisot and Salem scales

Henry Zweiman

September 9, 2026

Abstract

Let be multiplicatively independent real algebraic integers, each with all its other conjugates in the closed unit disk. For positive algebraic , we prove an effective upper bound for the product of the two Bernoulli Fourier products at arguments and . The result applies to Pisot–Pisot, Pisot–Salem, and Salem–Salem pairs. The proof gives a quantitative lower bound on the combined number of large fractional parts at the two scales. Its ingredients are a uniform algebraic projection on a near-integer interval, a height bound that excludes exact resonance, and the theorem on linear forms in logarithms. We also obtain the same decay rate for convolutions of homogeneous self-similar measures with algebraic translations.

1. Statement and context

All logarithms are natural. Write for distance to the nearest integer, and, for , put

The product converges: its tail factors differ from one by a summable sequence. A Pisot number is a real algebraic integer greater than one whose other conjugates have modulus less than one. Allowing conjugates on the unit circle also includes Salem numbers. Independence means , with , only when .

Theorem 1.

Let be multiplicatively independent real algebraic integers. Suppose all other conjugates of each have modulus at most one. Let be algebraic. There are effectively computable constants and , depending only on these four numbers, such that

The qualitative Pisot product problem has a classical solution in the nonexceptional parameter range: Senge and Straus [4], Theorem 2, give its multiplicative-independence criterion. The case has the elementary identity and must be excluded from an “if and only if” formulation; it causes no problem for the sufficient condition in Theorem 1. Stewart [5] gives effective joint digit estimates in integer bases. The present question is quantitative joint decay with arbitrary nonrational Pisot and Salem parameters and algebraic rescalings.

Nguyen [3], Theorem 1.6, proves qualitative decay for Pisot products with arguments having algebraic main terms. His Section 3 extracts algebraic approximants from long intervals of small fractional parts. We use a uniform projection version of this mechanism and make the interval ratio grow with the frequency. This permits an effective logarithm estimate despite the changing approximants.

For a single Salem parameter, Marshall-Maldonado and Solomyak [2], Corollary B.1, obtain an upper bound of the form . Our result concerns a product with independent parameters and does not improve their single-factor assertion. Varjú and Yu [6], Theorem 1.10, give a stronger joint fractional-part bound for one fixed base at two arguments whose ratio is not Liouville over the field of that base. Here the bases differ; the ratio of arguments may be one. No assertion of power Fourier decay, absolute continuity, or optimality of (1) is made.

2. Two height estimates

Use the absolute logarithmic Weil height , with . For a number field , normalize its absolute values so that the product formula holds and

We use , , and .

Lemma 2.

For fixed independent positive algebraic numbers , an effectively computable satisfies

Proof.

Take a field containing the two numbers and let consist of its archimedean places and the finite places where either number is not a unit. Their logarithm vectors are linearly independent over . Indeed, a rational dependence would make a nontrivial have height zero; this is a positive real root of unity, hence one. An irrational dependence would, by rational approximation, give infinitely many distinct such products with logarithm vectors bounded (in fact tending to zero). They have bounded degree and height, contrary to Northcott’s finiteness theorem.

Choose a nonzero two-by-two minor of the matrix with columns . The inverse matrix bounds by a constant times , proving the estimate. This also proves effectivity. The set and its logarithms are computable from algebraic data. Certified approximations will eventually identify and bound away from zero some nonzero minor, since one exists. An upper bound for the inverse norm then gives a positive lower bound for . No effective version of Northcott finiteness is required.

Lemma 3.

Fix positive algebraic and a number field containing them. There is an effective such that, for positive , , and with , one has

Proof.

Apply the multiplicative form of Matveev’s estimate, as stated in Bugeaud–Mignotte–Siksek [1], Theorem 9.4. For logarithms, the real-field bound is

where , , and . Take the numbers and exponents , omitting any term equal to one or having exponent zero. The parameters for are fixed. Since , the remaining parameter is at most a fixed multiple of . This gives (3); when only one term remains, the same estimate or the elementary height inequality suffices.

3. Projection from a near-integer interval

Lemma 4.

Fix a real algebraic integer of degree , all of whose other conjugates have modulus at most one. There are effective and with the following property. If , , , and

there is such that

Proof.

Write for the monic minimal polynomial. Choose and let be the nearest integer to . For , the integer has absolute value less than one, hence is zero.

Let be the distinct roots of . The initial values and the recurrence give unique coefficients with

The inverse of the fixed Vandermonde matrix has bounded entries. Since in this initial range, , with effective .

Write

This polynomial is one at and zero at the other roots, so . Set . The height inequalities, with the fixed and , give the first bound in (4). At the last index, (5) gives

This uses only , so it includes the Salem case. All constructions also apply when , with an empty final sum.

4. Independent scales force large fractional parts

For , define the finite count

Theorem 5.

Under the hypotheses of Theorem 1, there are effective and such that

Proof.

Write and , with , , and set . For effectively large , both exponents are positive, is comparable to , and is bounded above and below by positive constants. All constants below depend only on the fixed data. Take smaller than the two thresholds in Lemma 4 and take a fixed integer .

Choose

where the effective constant will be fixed sufficiently large. For an integer , put , . Consider only indices for which . Then and

We claim that at least one distance is at least on the paired intervals and for the sequences and , respectively. Suppose otherwise. Lemma 4 gives , with

For sufficiently large these errors are less than , so are positive and uniformly bounded above and away from zero. Set

Since , the preceding estimates and (7) imply, with fixed effective constants ,

Here fixed prefactors are absorbed by increasing the lower bound on ; comparability of is uniform in .

If , Lemma 2 gives

which is impossible for large . If , apply Lemma 3 in the fixed field . It gives , contradicting (8) once . This proves the claim.

There are sufficiently many disjoint pairs of intervals. Precisely, put

All are allowed. For effectively large , . For each base the intervals are disjoint, because , and similarly for . Their indices lie between zero and or , since . Reversing indices, , shows that every large distance supplied by the claim is counted in (6). Distinct pairs give distinct counted factors. Finally converts the lower bound for into (6). All thresholds used are effective.

Proof of Theorem 1.

At each factor counted in Theorem 5, . Every remaining factor has modulus at most one. Thus the product is bounded by , proving the theorem.

5. Homogeneous self-similar measures

Use the Fourier convention . A homogeneous self-similar probability measure with contraction and translations satisfies

Corollary 6.

Let satisfy Theorem 1. Let be homogeneous self-similar probability measures with these reciprocal contractions, positive probability vectors, and algebraic translations. Suppose each translation set has at least two distinct elements. Then satisfies the bound (1) with suitable . The constants are effective when the probabilities are also given as algebraic numbers.

Proof.

Choose distinct translations for and for , and put , . For a factor of the first Fourier product,

The analogous inequality holds for the second measure. Each distance counted in Theorem 5 therefore supplies a factor bounded by a fixed number strictly less than one. Multiplication gives the assertion, since . For arbitrary positive probabilities one can choose positive rational lower bounds for the selected weights to obtain existence of constants; algebraic input makes this choice algorithmic.

For a concrete mixed pair take and let be the largest root of . Indeed, ; its other conjugate lies in , giving two unit-circle roots. The polynomial is irreducible: has a negative conjugate in the real quadratic field, so is not a square there. Thus has degree four and is Salem. These parameters are independent. More generally, a number satisfying our conjugate assumption obeys for every : an embedding fixing must send to a conjugate of modulus , and there is only one. Hence two such numbers of different degrees cannot have a common positive power.

The loss of comes from intervals with ratio : the logarithm estimate charges a factor for exponents of size . Improving this rate requires additional input beyond the estimates used here. The theorem supplies no comparable assertion for a single Salem factor or for two multiplicatively dependent nonexceptional Pisot bases.

References

[1] Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), 969–1018. Authoritative full text.

[2] J. Marshall-Maldonado and B. Solomyak, Quantitative weak mixing for typical Salem substitution suspension flows, arXiv:2601.15035v1 (2026).

[3] K. D. Nguyen, Algebraic numbers and Fourier analysis: Salem’s third problem, arXiv:2604.18875v1 (2026).

[4] H. G. Senge and E. G. Straus, PV-numbers and sets of multiplicity, Period. Math. Hungar. 3 (1973), 93–100. doi:10.1007/BF02018464.

[5] C. L. Stewart, On the representation of an integer in two different bases, J. Reine Angew. Math. 319 (1980), 63–72. Author’s copy.

[6] P. P. Varjú and H. Yu, Fourier decay of self-similar measures and self-similar sets of uniqueness, arXiv:2004.09358v2 (2021).