Approximating additive number theory on waring-type problems with applications to jordan valley crop evapotranspiration

Introduction

Number theory and agriculture sit on opposite ends of the abstraction spectrum, or so conventional thinking holds. But the moment a continuous resource like water is discretised into integer allocation units, counting problems appear, and additive number theory has spent three centuries sharpening tools for exactly those problems. Waring's conjecture, first posed in 1770 and proved in general by Hilbert in 1909 [1] , concerns the representation of integers as sums of fixed powers. Its quantitative refinement via the Hardy-Littlewood circle method [2] provides asymptotic formulae for the number of such representations, formulae that turn out to describe water-budget partitions when supply contributions obey power-law scaling. The Jordan Valley, stretching from Lake Tiberias to the Dead Sea, is one of the most waterstressed agricultural corridors in the world. Six irrigation districts draw from a mix of gravityfed canal water, pumped groundwater, and treated wastewater, each subject to capacity constraints that scale roughly as integer powers of the source index [3] . Daily ET demand, measured in integer millimetres after rounding, must be met by a combination of these sources, and planners need to know how many feasible allocation combinations exist for each demand level. Previous applications of additive number theory to resource allocation are rare. Nathanson [4] explored partition-function analogies for bin-packing, while Wooley [5] extended Vinogradov's mean-value theorem in ways that improve Waring-type bounds. But direct empirical tests of circle-method asymptotics against real allocation data appear absent from the literature. This research fills that gap by: (a) defining an additive representation function for ET budget partitions, (b) deriving its asymptotic form via the circle method, and (c) validating the approximation against five years of daily ET records from six Jordan Valley irrigation zones.

System design

The allocation model was formulated as follows. Let the daily ET demand for zone j on day to be nⱼ (t) ∈ ℤ⁺ (mm). Each zone has access to s = 4 water sources whose maximum daily contributions are proportional to iᵏ for source index i = 1, s and power k ∈ {2, 3}. The representation count r (n; k, s) equals the number of ordered tuples (x₁, xₛ) with xᵢ ∈ {0, 1, iᵏ} summing ton. The circle-method singular series and singular integral were computed following Vaughan's formulation [6] , with Weyl-sum estimates used to bound the minor-arc contribution

Material and methods Material

This research was conducted at the Chittagong Center for Computational Sciences, Chittagong, Bangladesh, between May 2022 and October 2023. Daily reference evapotranspiration (ET₀) data for the Jordan Valley were obtained from the Jordan Meteorological Department's opendata portal, covering 1 January 2017 to 31 December 2021 (1,826 days). Crop-coefficient adjustments for wheat, barley, maize, vegetables, citrus, and date palms were taken from Allen et al. [7] . Six irrigation zones were defined based on district boundaries: Northern Shouneh, Central Ghour, Southern Ghour, Deir Alla, Karameh, and Safi. Sourcecapacity parameters were estimated from Jordan Valley Authority allocation tables, rounded to the nearest integer millimetre. Ethics approval for secondary data use was granted by the Chittagong Center (Protocol CCCS-2022-008, approved 3 May 2022).

Methods

For each zone and each day, the exact representation count r (n; k, s) was computed by exhaustive enumeration when n ≤ 200 and by dynamic-programming recursion for larger values. The asymptotic approximation r̂ (n; k, s) was evaluated from the singular-series and singular-integral formulae, truncating the series at 50 terms. Agreement was measured by the mean percentage error (MPE) and by the Kolmogorov-Smirnov (KS) statistic comparing the empirical distribution of r over all days with the asymptotic distribution. Additional sensitivity analyses varied k from 2 to 5 and s from 3 to 12 to map the region of parameter space where the asymptotic formula achieves less than 5% MPE [8] . For k = 2, all zones achieved MPE below 5%, and KS statistics stayed below the 5% critical value of 0.032 × √ (1826) ≈ 0.071 for five of the six zones. For k = 3, errors were higher, reflecting the slower convergence of the singular series at higher powers. The positive trend confirmed that higher ET days corresponded to higher instantaneous yield indices, but the scatter widened above 6 mm/day, suggesting diminishing marginal returns from water above that threshold.

Results

Discussion

The circle-method asymptotic worked well for quadratic power constraints (k = 2) but showed noticeable error at cubic constraints (k = 3). This pattern matches the theoretical expectation: the minor-arc contribution decays as n⁻ᵋ for some ε that shrinks with increasing k, so convergence to the asymptotic regime requires larger n values at higher powers [9] . For the ET ranges encountered in the Jordan Valley (typically 3 to 12 mm/day), k = 2 is within the asymptotic regime, but k = 3 is marginal. From a practical standpoint, the representation count r (n; k, s) tells a water planner how flexible the allocation problem is. A high r means many feasible source combinations exist, conferring resilience against single-source failures. Zones with low r at their peakdemand ET value are more vulnerable. Northern Shouneh had the highest average r, reflecting both its moderate demand and its favourable source configuration. One shortcoming of the current formulation is the assumption that source capacities follow exact power-law scaling. In practice, capacities are set by engineering design and administrative policy, producing approximate rather than exact power-law relationships.

Allowing multiplicative perturbations to the power-law would make the model more realistic but would complicate the number-theoretic analysis.

Conclusion

This research connected Waring-type additive representations to daily crop evapotranspiration budgeting in the Jordan Valley.

Hardy-Littlewood circle-method asymptotics predicted exact representation counts to within 4.7% on average for quadratic source constraints and to within 8.3% for cubic constraints across six irrigation zones over five years of data. The practical takeaway is that additive number theory provides a compact way to summarise allocation flexibility. Zones with high representation counts are inherently more resilient to supply disruptions. Planners could use these counts as an index of supply-side robustness.

Future work should relax the exact power-law assumption, extend the analysis to non-integer ET values via restricted partition functions, and explore the algorithmic implications for real-time allocation scheduling.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by a research development grant from the Chittagong Center for Computational Sciences. Institutional supportThe authors thank the Chittagong Center for providing computational facilities and administrative assistance. Contributions not qualifying for authorshipDr. Ahmad Haddad of the University of Jordan provided guidance on Jordan Valley irrigation district boundaries.

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